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diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md new file mode 100644 index 0000000..857d298 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md @@ -0,0 +1,38 @@ +<h1><div align=”center”><b>SubTopic: Critical Points</b></h1></div> +<br/></br> + +<tab>file1_Critical_Point_of_a_function + + +<br/></br> +<br/></br> + +<tab>file2_Traces_and_Tangent + + +<br/></br> +<br/></br> + +<tab>file3_Tangent_plane_at_extrema_of_a_function + + +<br/></br> +<br/></br> + +<tab>file4_Relative_Maximum_and_Relative_Minimum + + +<br/></br> +<br/></br> + +<tab>file5_Saddle_Point + + +<br/></br> +<br/></br> + +<tab>file6_f(x,y)=(y-x)(1-2x-3y) + +%3D(y-x)(1-2x-3y).gif?raw=true) +<br/></br> +<br/></br> diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.gif Binary files differnew file mode 100644 index 0000000..ca3989c --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.py new file mode 100644 index 0000000..e8cb08d --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.py @@ -0,0 +1,77 @@ +from manimlib.imports import* +import math as m + +#---- case 1: parial derivatives exist at critical point of the function +class firstScene(ThreeDScene): + def construct(self): + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #---- f(x,y) = e^(-10x^2-10y^2) + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + m.exp(-10*u**2-10*v**2) + ]),u_min = -1, u_max = 1, v_min = -1, v_max = 1, checkerboard_colors = [TEAL_E,TEAL_D,TEAL_C,TEAL_B]).fade(0.6).scale(3.5).shift([0,0,1.5]) + + l1 = Line([0,0,3.75],[0,0,0],color = '#800000') + + d = Dot([0,0,3.75],color = '#800000') #---- critical point + + d_text = TextMobject("$\\frac{\\partial f}{\\partial x}=\\frac{\\partial f}{\\partial y} = 0$").scale(0.8).to_corner(UL) + + f_text = TextMobject("Critical Point ",color = YELLOW).shift(3.4*UP).scale(0.5) + + self.set_camera_orientation(phi = 45*DEGREES, theta = 40*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.add_fixed_in_frame_mobjects(d_text) + self.begin_ambient_camera_rotation(rate = 0.2) + self.play(Write(surface)) + self.wait(1) + self.play(Write(l1)) + self.play(Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(f_text) + self.wait(3) + self.play(FadeOut(f_text),FadeOut(surface),FadeOut(axes),FadeOut(d_text),FadeOut(d),FadeOut(l1),FadeOut(label_x),FadeOut(label_y)) + + +#---- case 2: parial derivatives do not exist at critical point of the function +class secondScene(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #---- g(x,y)= |x|+|y| + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + abs(u)+abs(v) + ]),u_min = -1.5, u_max = 1.5, v_min = -1.5, v_max = 1.5, checkerboard_colors = [TEAL_E,TEAL_D,TEAL_C,TEAL_B]) + + d2 = Dot([0,0,0],color = '#800000') #---- critical point + + d2_text = TextMobject("$\\frac{\\partial f}{\\partial x}$ and/or $\\frac{\\partial f}{\\partial y}$ does not exist").scale(0.7).to_corner(UL) + + g_text = TextMobject("Critical Point",color = YELLOW).shift(1.2*RIGHT).scale(0.6) + + self.set_camera_orientation(phi = 60*DEGREES, theta = 40*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.add_fixed_in_frame_mobjects(d2_text) + self.begin_ambient_camera_rotation(rate = 0.2) + self.wait(1) + self.play(Write(surface2)) + self.wait(1) + self.play(Write(d2)) + self.wait(1) + self.add_fixed_in_frame_mobjects(g_text) + self.wait(2) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif Binary files differnew file mode 100644 index 0000000..84acf2e --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.py new file mode 100644 index 0000000..4b020e1 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.py @@ -0,0 +1,88 @@ +from manimlib.imports import* +import math as m + +#---- tangent to the trace with x constant +class firstScene(ThreeDScene): + def construct(self): + + axes = ThreeDAxes().scale(1) + label_x = TextMobject("$x$").shift([5.8,-0.5,0]) + label_y = TextMobject("$y$").shift([-0.5,-5.6,0]).rotate(-4.5) + + #---- graph of f(x,y) = -x^2-y^2 + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2-v**2 + ]),u_min=-1,u_max=1, v_min=-1,v_max=1,checkerboard_colors=[PURPLE_C,PURPLE_D,PURPLE_E,PURPLE_B]).scale(1.5).shift([0,0,2]).rotate(0.2) + + #---- curve(trace) along y axis + curve = ParametricSurface( + lambda u, v: np.array([ + u*0.4, + v, + -v**2 + ]),v_min =-1 , v_max =1 , u_min = -0.1, u_max = 0.1).scale(1.6).shift([0.02,0.1,2.3]).set_color("#800000").rotate(0.1) + + d = Dot(color =YELLOW).shift([-0.05,-0.2,2.3]) #---- critical point + + x_text = TextMobject("Tangent to the trace with $x$ constant at critical point").shift(3*RIGHT+2*UP).scale(0.5).to_corner(UL) + + tangent_line = Line([-0.05,-1.5,2.3],[-0.05,1.5,2.3],color = '#228B22') + + self.add(axes) + self.set_camera_orientation(phi = 40 * DEGREES, theta = 55 * DEGREES) + self.begin_ambient_camera_rotation(rate = 0.1) + self.add(label_x) + self.add(label_y) + self.play(Write(surface)) + self.add_fixed_in_frame_mobjects(x_text) + self.add(curve) + self.wait(1) + self.play(Write(tangent_line),Write(d)) + self.wait(1) + + + +#---- tangent to the trace with y constant +class secondScene(ThreeDScene): + def construct(self): + + axes = ThreeDAxes().scale(1) + label_x = TextMobject("$x$").shift([5.8,-0.5,0]) + label_y = TextMobject("$y$").shift([-0.5,-5.6,0]).rotate(-4.5) + + #---- graph of f(x,y) = -x^2-y^2 + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2-v**2 + ]),u_min = -1, u_max = 1, v_min = -1, v_max = 1, checkerboard_colors = [PURPLE_B,PURPLE_C,PURPLE_D,PURPLE_E]).scale(1.5).shift([0,0,2]).rotate(0.2) + + #---- curve(trace) along x axis + curve = ParametricSurface( + lambda u, v: np.array([ + u, + v*0.4, + -u**2 + ]),v_min = -0.1, v_max = 0.1, u_min = -1, u_max = 1).scale(1.6).shift([0.07,0.1,2.3]).set_color("#800000") + + d = Dot(color = YELLOW).shift([0,-0.2,2.3]) #---- critical point + + tangent_line = Line(color = '#228B22').scale(1).shift([0,-0.2,2.3]).rotate(m.radians(190),LEFT) + + y_text = TextMobject("Tangent to the trace with $y$ constant at critical point").shift(3*RIGHT+2*UP).scale(0.5).to_corner(UL) + + self.add(axes) + self.set_camera_orientation(phi = 40 * DEGREES, theta = 55 * DEGREES) + self.add(label_x) + self.add(label_y) + self.begin_ambient_camera_rotation(rate = 0.1) + self.play(Write(surface)) + self.add_fixed_in_frame_mobjects(y_text) + self.add(curve) + self.wait(1.5) + self.play(Write(tangent_line),Write(d)) + self.wait(0.5) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif Binary files differnew file mode 100644 index 0000000..14fb318 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.py new file mode 100644 index 0000000..e674113 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.py @@ -0,0 +1,73 @@ +from manimlib.imports import* + +#---- tangent plane to minima of the function +class firstScene(ThreeDScene): + def construct(self): + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #---- parabola: f(x,y) = x**2 + y**2 + parabola = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2+v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [GREEN_E,GREEN_D,GREEN_C,GREEN_B], resolution = (20, 20)).scale(1) + + d = Dot(np.array([0,0,0]), color = '#800000') # ---- critical point + + tangent_plane = Rectangle(fill_color = '#C0C0C0', fill_opacity = 0.3).move_to(ORIGIN).fade(0.7) # ----tangent plane + + parabola_text = TextMobject("Minimum with horizontal tangent plane").scale(0.7).to_corner(UL) + + self.set_camera_orientation(phi = 75 * DEGREES, theta = 45 * DEGREES) + self.begin_ambient_camera_rotation(rate = 0.2) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.add_fixed_in_frame_mobjects(parabola_text) + self.wait(1) + self.play(Write(parabola)) + self.play(ShowCreation(d)) + self.wait(1) + self.play(ShowCreation(tangent_plane)) + self.wait(2) + self.play(FadeOut(parabola_text),FadeOut(parabola),FadeOut(tangent_plane),FadeOut(d),FadeOut(label_x),FadeOut(label_y),FadeOut(axes)) + + +#---- tangent plane to maxima of the function +class secondScene(ThreeDScene): + def construct(self): + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #----parabola: g(x,y) = -x**2-y**2 + parabola = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2-v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [BLUE_E,BLUE_D,BLUE_C,BLUE_B], resolution = (20, 20)).scale(1) + + d = Dot(np.array([0,0,0]), color = '#800000') #---- critical point + + tangent_plane = Rectangle(fill_color = '#C0C0C0',fill_opacity = 0.3).move_to(ORIGIN).fade(0.7) #---- tangent plane + + parabola_text = TextMobject("Maximum with horizontal tangent plane").scale(0.7).to_corner(UL) + + self.set_camera_orientation(phi = 75 * DEGREES, theta = 45 * DEGREES) + self.begin_ambient_camera_rotation(rate = 0.2) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.add_fixed_in_frame_mobjects(parabola_text) + self.wait(1) + self.play(Write(parabola)) + self.play(ShowCreation(d)) + self.wait(1) + self.play(ShowCreation(tangent_plane)) + self.wait(2) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.gif Binary files differnew file mode 100644 index 0000000..6b93359 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.py new file mode 100644 index 0000000..3bd810d --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.py @@ -0,0 +1,51 @@ +from manimlib.imports import* +import math as m + +#---- locating extrema of a funtion using critical points +class Extrema(ThreeDScene): + def construct(self): + + h_text = TextMobject("Relative Maximum and Relative Minimum",color = GREEN) + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis + + #---- f(x,y) = 5(x+y)e^(-x^2-y^2) + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + 5*(u+v)*m.exp(-u**2-v**2) + ]),u_min = -PI, u_max = PI, v_min = -PI, v_max = PI).set_color(TEAL).shift([0,0,0]).fade(0.4) + + d1 = Dot(color = YELLOW).shift([0.5,0.5,3.02]) #---- critical point for maxima + l1 = Line([0.5,0.5,0.1],[0.5,0.5,3],color = YELLOW) + + d2 = Dot(color = YELLOW).shift([-1.15,0,-2.98]) #---- critical point for minima + l2 = Line([-1.15,0,0],[-1.15,0,-2.98],color = YELLOW) + + max_text = TextMobject("Relative Maximum").shift(3.1*UP+1.5*RIGHT).scale(0.5) + min_text = TextMobject("Relative Minimum").shift(3.1*DOWN+1.5*LEFT).scale(0.5) + + self.add_fixed_in_frame_mobjects(h_text) + self.wait(1) + self.wait(1) + self.play(FadeOut(h_text)) + self.wait(1) + self.set_camera_orientation(phi = 100*DEGREES, theta = -40*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(surface)) + self.wait(1) + self.play(Write(l1),Write(d1)) + self.add_fixed_in_frame_mobjects(max_text) + self.wait(1) + self.play(Write(l2),Write(d2)) + self.add_fixed_in_frame_mobjects(min_text) + self.wait(1) + self.wait(1) + self.play(FadeOut(l1),FadeOut(d1),FadeOut(l2),FadeOut(d2),FadeOut(max_text),FadeOut(min_text)) + self.begin_ambient_camera_rotation(rate = 0.3) + self.wait(3) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.gif Binary files differnew file mode 100644 index 0000000..7300f3a --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.py new file mode 100644 index 0000000..67dbb18 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.py @@ -0,0 +1,71 @@ +from manimlib.imports import* +import math as m + +#---- saddle point of a function +class SaddlePoint(ThreeDScene): + def construct(self): + + h_text = TextMobject("Saddle Point",color = GREEN) + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis + + #---- f(x,y) = -x^2-y^2 + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2-v**2 + ]),u_min = -1, u_max = 1, v_min = -1, v_max = 1,checkerboard_colors = [BLUE_B,BLUE_C,BLUE_D,BLUE_E]).shift([0,0,0]).scale(3) + + #---- curve(trace) along y axis + curve_x = ParametricSurface( + lambda u, v: np.array([ + u*0.1, + v, + v**2 + ]),v_min = -1, v_max = 1, u_min = -0.2, u_max = 0.2).shift([0,0,-2]).scale(3.1).set_color("#800000").rotate(m.radians(180),UP) + + x_text = TextMobject("A dip at critical point along x axis").scale(0.5).to_corner(UL) + + #---- curve(trace) along x axis + curve_y = ParametricSurface( + lambda u, v: np.array([ + u, + v*0.1, + -u**2 + ]),v_min = -0.2, v_max = 0.2, u_min = -1, u_max = 1).scale(3).shift([0.1,0,2.2]).set_color("#800000").rotate(m.radians(182),DOWN) + + y_text = TextMobject("A peak at critical point along y axis").scale(0.5).to_corner(UL) + + d = Dot(color = YELLOW).shift([0,-0.22,0]) #---- critical point(saddle point) + + self.add_fixed_in_frame_mobjects(h_text) + self.wait(1) + self.play(FadeOut(h_text)) + self.wait(1) + self.set_camera_orientation(phi = 75*DEGREES, theta = 40*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(surface)) + self.wait(1) + self.move_camera(phi = 45*DEGREES, theta = 70*DEGREES) + self.add(curve_y) + self.play(Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(x_text) + self.wait(1) + self.wait(1) + self.play(FadeOut(curve_y),FadeOut(d),FadeOut(x_text)) + self.wait(1) + self.move_camera(phi = 40*DEGREES, theta = 30*DEGREES) + self.add(curve_x) + self.play(Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(y_text) + self.begin_ambient_camera_rotation(rate = 0.3) + self.wait(3) + self.play(FadeOut(curve_x),FadeOut(d),FadeOut(y_text)) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).gif Binary files differnew file mode 100644 index 0000000..4bc92f8 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).py new file mode 100644 index 0000000..41c3b61 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).py @@ -0,0 +1,29 @@ +from manimlib.imports import* + +#---- visualization of the function +class ExampleAnimation(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #---- f(x,y) = (y-x)(1-2x-3y) + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + (v-u)*(1-2*u-3*v) + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [PURPLE_B,PURPLE_C,PURPLE_D, PURPLE_E]).scale(1).fade(0.2).shift([0.2,0.2,0]) + + f_text = TextMobject("$f(x,y) = (y-x)(1-2x-3y)$").to_corner(UL) + + self.set_camera_orientation(phi = 60 * DEGREES, theta = 75 * DEGREES) + self.begin_ambient_camera_rotation(rate=0.1) + self.add_fixed_in_frame_mobjects(f_text) + self.wait(1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.wait(1) + self.play(Write(f)) + self.wait(4) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.gif Binary files differnew file mode 100644 index 0000000..9d64d50 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.py new file mode 100644 index 0000000..da17aac --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.py @@ -0,0 +1,57 @@ +from manimlib.imports import* +import math as m + +#---- optimizing funtion f(x,y) w.r.t to g(x,y) +class ConstrainedExtrema(ThreeDScene): + def construct(self): + axes = ThreeDAxes().fade(0.4) + label_x = TextMobject("$x$").shift([5.5,-0.5,0]).fade(0.4) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5).fade(0.4) #---- y axis + + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2+v**2+u**3-v**3 + ]),u_min=-0.5,u_max=0.5, v_min=-0.5,v_max=0.5).scale(5).shift([0,1,2.5]).set_color(TEAL).fade(0.4) + + c = Circle().set_color('#FF00FF').shift([-0.4,0,1.5]).rotate(1.9,UP).scale(0.7) + + minima = Dot(color = '#4169E1').shift([-0.5,0.5,1]).rotate(1.571,UP) + maxima = Dot(color = '#4169E1').shift([0.1,0,2.2]).rotate(1.571,UP) + + l1 = DashedLine([-0.5,0.5,0.9],[-0.5,0.5,0],color = '#F08080') + l2 = DashedLine([0.1,0,2.1],[0.1,0,0],color = '#F08080') + + c2 = Circle(fill_opacity= 0.5).shift([-0.3,0.2,0]).scale(0.4) + + minima_refl = Dot(color = '#4682B4').shift([-0.5,0.5,0]).rotate(1.571,UP) + maxima_refl = Dot(color = '#4682B4').shift([0.1,0,0]).rotate(1.571,UP) + + max_text = TextMobject("maximum over $g(x,y)=k$",color = '#FFA074').shift([-1.7,0,0]).scale(0.5).shift(2.2*UP) + min_text = TextMobject("minimum over $g(x,y)=k$",color = '#FFA074').shift([2.5,0.5,1]).scale(0.5).shift(0.5*UP) + label_f = TextMobject("$z=f(x,y)$",color = '#8A2BE2').scale(0.5).shift(3*UP+3*RIGHT) + label_g = TextMobject("$g(x,y)=k$",color = '#8A2BE2').scale(0.5).shift(2*RIGHT) + + self.add(axes) + self.add(label_x) + self.add(label_y) + self.set_camera_orientation(phi=75*DEGREES,theta=45*DEGREES) + self.play(Write(surface)) + self.add_fixed_in_frame_mobjects(label_f) + self.wait(2) + self.play(Write(c)) + self.wait(1) + self.play(Write(maxima)) + self.add_fixed_in_frame_mobjects(max_text) + self.wait(1) + self.play(Write(minima)) + self.add_fixed_in_frame_mobjects(min_text) + self.wait(1) + self.play(ShowCreation(l1),ShowCreation(l2)) + self.play(Write(c2)) + self.add_fixed_in_frame_mobjects(label_g) + self.wait(1) + self.play(Write(maxima_refl)) + self.play(Write(minima_refl)) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file2_Geometric_Proof.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file2_Geometric_Proof.py new file mode 100644 index 0000000..2c2a9de --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file2_Geometric_Proof.py @@ -0,0 +1,90 @@ +from manimlib.imports import* + +#---- visualization of geometric proof of Lagrange multiplier +class GeometricProof(ThreeDScene): + def construct(self): + axes = ThreeDAxes().scale(0.7).rotate(math.radians(180)) + label_x = TextMobject("$x$").shift(4*RIGHT).fade(0.4) #---- x axis + label_y = TextMobject("$y$").shift(3.2*DOWN+0.2*RIGHT).rotate(math.radians(180)).fade(0.4) #---- y axis + + surface = ParametricSurface( + lambda u, v: np.array([ + 1*np.sin(u)*np.cos(v), + 1*np.sin(u)*np.sin(v), + -1*np.sin(u)*np.sin(u)+2 + ]),u_min=0,u_max=PI/2,v_min=0,v_max=2*PI,checkerboard_colors=[GREEN_C, GREEN_E]).scale(1).shift([-1.5,-1.5,0]) + + d = Dot([-2,-2.55,0],color = '#800000') + a_df = Arrow(color = '#00FFFF').rotate(-2).shift(3.2*DOWN+2.3*LEFT) #---- f parallel to g at maxima + a_dg = Arrow(color = '#FF00FF').scale(0.8).shift(3.2*DOWN+2.3*LEFT).rotate(-2) #---- g parallel to f at maxima + + b_dg = Arrow(color = '#00FFFF').rotate(1.1).shift(0.82*LEFT+0.15*UP) #---- g parallel to f at minima + b_df = Arrow(color = '#FF00FF').scale(0.6).rotate(-2).shift(1.43*LEFT+1.1*DOWN) #---- f parallel to g at minima + + + qd = Dot(color = '#800000').shift(1.2*LEFT+0.6*DOWN) + + #---- level curves + l1 = Line([-1,-3.1,0],[-4,-3.1,0],color = PINK).rotate(-0.3).fade(0.6) + l2 = Line([-0.9,-2.9,0],[-4,-2.9,0],color = PINK).rotate(-0.3).fade(0.6) + l3= Line([-0.8,-2.7,0],[-4,-2.7,0],color = PINK).rotate(-0.3).fade(0.6) + l4= Line([-0.7,-2.45,0],[-4,-2.45,0],color = PINK).rotate(-0.3).fade(0.6) + l5= Line([-0.6,-2.2,0],[-4,-2.25,0],color = PINK).rotate(-0.3).fade(0.6) + l6 = Line([-0.5,-2,0],[-4,-2,0],color = PINK).rotate(-0.3).fade(0.6) + l7 = Line([-0.4,-1.8,0],[-4,-1.8,0],color = PINK).rotate(-0.3).fade(0.6) + l8 = Line([-0.3,-1.6,0],[-4,-1.6,0],color = PINK).rotate(-0.3).fade(0.6) + l9= Line([-0.2,-1.4,0],[-4,-1.4,0],color = PINK).rotate(-0.3).fade(0.6) + l10= Line([-0.1,-1.2,0],[-4,-1.2,0],color = PINK).rotate(-0.3).fade(0.6) + l11 = Line([-0,-1,0],[-4,-1,0],color = PINK).rotate(-0.3).fade(0.6) + l12 = Line([-0,-0.8,0],[-4,-0.8,0],color = PINK).rotate(-0.3).fade(0.6) + l13= Line([-0,-0.55,0],[-4,-0.55,0],color = PINK).rotate(-0.3).fade(0.6) + l14= Line([-0,-0.35,0],[-4,-0.35,0],color = PINK).rotate(-0.3).fade(0.6) + l15= Line([-0.,-0.15,0],[-4,-0.15,0],color = PINK).rotate(-0.3).fade(0.6) + + rel_text = TextMobject("$\\nabla f = \\lambda \\nabla g$",color = TEAL).shift([3,3.2,0]).scale(0.5) + + f_text = TextMobject("$\\nabla f$",color = '#800000').shift([1,1,0]).scale(0.5) + g_text = TextMobject("$\\nabla g$").shift([1.2,-0.8,0]).scale(0.5) + + p_text= TextMobject("$P$").shift([1.8,2.6,0]).scale(0.5) + + #---- labelling of level curves + l1_text = TextMobject("$w=$ 17").rotate(math.radians(180)).scale(0.4).shift(2.7*DOWN+4.36*LEFT) + l2_text = TextMobject("$w=$ 16").rotate(math.radians(180)).scale(0.4).shift(2.46*DOWN+4.36*LEFT) + l3_text = TextMobject("$w=$ 15").rotate(math.radians(180)).scale(0.4).shift(2.2*DOWN+4.36*LEFT) + l4_text = TextMobject("$w=$ 14").rotate(math.radians(180)).scale(0.4).shift(1.97*DOWN+4.36*LEFT) + l5_text = TextMobject("$w=$ 13").rotate(math.radians(180)).scale(0.4).shift(1.74*DOWN+4.36*LEFT) + l6_text = TextMobject("$w=$ 12").rotate(math.radians(180)).scale(0.4).shift(1.5*DOWN+4.36*LEFT) + l7_text = TextMobject("$w=$ 11").rotate(math.radians(180)).scale(0.4).shift(1.26*DOWN+4.36*LEFT) + l8_text = TextMobject("$w=$ 10").rotate(math.radians(180)).scale(0.4).shift(1.05*DOWN+4.36*LEFT) + l9_text = TextMobject("$w=$ 9").rotate(math.radians(180)).scale(0.4).shift(0.8*DOWN+4.32*LEFT) + l10_text = TextMobject("$w=$ 8").rotate(math.radians(180)).scale(0.4).shift(0.6*DOWN+4.32*LEFT) + l11_text = TextMobject("$w=$ 7").rotate(math.radians(180)).scale(0.4).shift(0.4*DOWN+4.32*LEFT) + l12_text = TextMobject("$w=$ 6").rotate(math.radians(180)).scale(0.4).shift(0.2*DOWN+4.32*LEFT) + l13_text = TextMobject("$w=$ 5").rotate(math.radians(180)).scale(0.4).shift(-0.02*DOWN+4.32*LEFT) + l14_text = TextMobject("$w=$ 4").rotate(math.radians(180)).scale(0.4).shift(-0.23*DOWN+4.32*LEFT) + l15_text = TextMobject("$w=$ 3").rotate(math.radians(180)).scale(0.4).shift(-0.44*DOWN+4.32*LEFT) + + level_Curve = VGroup(l1,l1_text,l2,l2_text,l3,l3_text,l4,l4_text,l5,l5_text,l6,l6_text,l7,l7_text,l8,l8_text,l9,l9_text,l10,l10_text,l11,l11_text,l12,l12_text,l13,l13_text,l14,l14_text,l15,l15_text) + + self.set_camera_orientation(phi=0 * DEGREES, theta = 90*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(surface)) + self.wait(1) + self.play(ShowCreation(level_Curve)) + self.wait(1) + self.play(ShowCreation(a_df),ShowCreation(a_dg),Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(rel_text) + self.add_fixed_in_frame_mobjects(p_text) + self.wait(1) + self.play(Write(qd)) + self.wait(1) + self.play(ShowCreation(b_df)) + self.add_fixed_in_frame_mobjects(f_text) + self.wait(1) + self.play(ShowCreation(b_dg)) + self.add_fixed_in_frame_mobjects(g_text) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.gif Binary files differnew file mode 100644 index 0000000..9602283 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.py new file mode 100644 index 0000000..bf75dd8 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.py @@ -0,0 +1,29 @@ +from manimlib.imports import* + +class firstScene(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + cylinder = ParametricSurface( + lambda u, v: np.array([ + np.cos(TAU * v), + np.sin(TAU * v), + 2 * (u) + ]),checkerboard_colors=[YELLOW_C,YELLOW_D,YELLOW_E] + ).fade(0.4) #Resolution of the surfaces + + plane = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u+v + ]),checkerboard_colors=[TEAL_C,TEAL_D,TEAL_E] + ).scale(2.5) + self.add(axes) + self.set_camera_orientation(phi=75*DEGREES,theta=45*DEGREES) + self.play(Write(cylinder)) + self.play(Write(plane)) + self.wait(1) + self.begin_ambient_camera_rotation(rate=0.7) + self.wait(5) + self.move_camera(phi=35*DEGREES,theta=-45*DEGREES) + self.wait(2) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.gif Binary files differnew file mode 100644 index 0000000..2b8bf5f --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.py new file mode 100644 index 0000000..4c17f90 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.py @@ -0,0 +1,50 @@ +from manimlib.imports import* + +#---- tangent plane is parallel to the surface of the funtion at a point +class firstScene(ThreeDScene): + def construct(self): + + s1_text=TextMobject("Suppose, the point $(x,y)$ lies on the surface of the function.").scale(0.5).shift(2*UP) + s2_text=TextMobject("When zooming on that point, the surface would appear more and more like a plane.").scale(0.5).shift(1*UP) + s3_text=TextMobject("This plane is called the tangent plane.").scale(0.5) + + #---- graph of function f(x,y) = -x^2-y^2 + + f = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2-v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [YELLOW_B,YELLOW_C,YELLOW_D, YELLOW_E]).shift([0,0,0]).scale(1) + + + d = Dot([0,0,0],color = '#800000') #---- critical point + + r = Rectangle(color = PURPLE,fill_opacity=0.2).shift([0.1,0,0]).scale(0.3) #---- tangent plane + + s = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2-v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [YELLOW_B,YELLOW_C,YELLOW_D, YELLOW_E]).shift([0,0,0]).scale(3.5) + + d2 = Dot([0,0,2.5],color = '#800000') #---- changing position of critical point + + r2 = Rectangle(color = PURPLE,fill_opacity=0.5).shift([0.1,0,2.5]).scale(0.3) #---- changing position of tangent plane + + self.set_camera_orientation(phi = 50 * DEGREES, theta = 45 * DEGREES) + self.add_fixed_in_frame_mobjects(s1_text) + self.add_fixed_in_frame_mobjects(s2_text) + self.add_fixed_in_frame_mobjects(s3_text) + self.wait(2) + self.play(FadeOut(s1_text)) + self.play(FadeOut(s2_text)) + self.play(FadeOut(s3_text)) + self.wait(1) + self.play(Write(f)) + self.play(Write(d)) + self.play(Write(r)) + self.wait(2) + self.play(ReplacementTransform(f,s),ReplacementTransform(d,d2),ReplacementTransform(r,r2)) + self.wait(2) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file2_Tangent_plane_approximation_visualization.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file2_Tangent_plane_approximation_visualization.gif Binary files differnew file mode 100644 index 0000000..d23405d --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file2_Tangent_plane_approximation_visualization.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file2_Tangent_plane_approximation_visualization.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file2_Tangent_plane_approximation_visualization.py new file mode 100644 index 0000000..d1ecf8c --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file2_Tangent_plane_approximation_visualization.py @@ -0,0 +1,74 @@ +from manimlib.imports import* + +#---- tangent plane is parallel to the x-y plane +class MaximaScene(ThreeDScene): + def construct(self): + + axes = ThreeDAxes().scale(1.2) + label_x= TextMobject("$x$").shift([5.4,-0.5,0]) #---- x axis + label_y= TextMobject("$y$").shift([-0.5,5.2,0]).rotate(-4.5) #---- y axis + + #---- graph of the function + s = ParametricSurface( + lambda u, v: np.array([ + 1.5*np.cos(u)*np.cos(v), + 1.5*np.cos(u)*np.sin(v), + 1.5*np.sin(u) + ]),u_min=0,u_max=PI,v_min=PI,v_max=2*PI,checkerboard_colors=[BLUE_B,BLUE_C,BLUE_D,BLUE_E]).scale(1.5).shift([-0.8,0.5,1.5]) + + d1 = Dot([0.2,2.01,2.24],color = '#800000').rotate(1.1,LEFT) #---- point(x_0,y_0) + d1_copy = Dot([1.1,2.2,-0.45],color = '#800000') #---- projection of point(x_0,y_0) on x-y plane + d1_text = TextMobject("$(x_0,y_0)$",color = "#8B0000").scale(0.4).shift(1.3*RIGHT+1.1*UP) + + d2 = Dot([1.1,2.2,2.7],color = '#800000').rotate(1,LEFT) #---- point(x,y) + d2_copy = Dot([0.1,1.95,0.4],color = '#800000') #---- projection of point(x,y) on x-y plane + d2_text = TextMobject("$(x,y)$",color = "#8B0000").scale(0.4).shift(0.6*RIGHT+0.8*UP) + + t_plane = Rectangle(color = PURPLE, fill_opacity=0.3).scale(0.4).rotate(1,LEFT).shift([1.1,2.5,2.9]) #---- tangent plane + + t_text= TextMobject("Tangent Plane",color = RED).scale(0.5).shift(0.3*RIGHT+1.3*UP).rotate(math.radians(5),LEFT) + + l1 = Line([1.1,2.2,2.6],[1.1,2.2,-0.45]).fade(0.2) + l2 = Line([0.1,1.95,2.05],[0.1,1.95,0.4]).fade(0.2) + + a1 = Line([0.1,1.95,0.4],[1.1,2.2,-0.45],color ="#00FF7F") + a_x = Line([0.1,1.95,0.4],[1.7,1.95,0.4],color ="#9400D3") + a_y = Line([0.1,1.95,0.4],[0.1,2.75,0.4],color ="#8B4513") + a2 = Line([1.7,1.95,0.4],[1.7,2.75,0.4]) + a3 = Line([0.1,2.75,0.4],[1.7,2.75,0.4]) + + #---- transition of tangent plane + + t2_plane = Rectangle(color = PURPLE, fill_opacity=0.3).scale(0.4).rotate(1,LEFT).shift([1.1,2.5,2]) + t3_plane = Rectangle(color = PURPLE, fill_opacity=0.3).scale(0.4).rotate(math.radians(180),LEFT).shift([1.1,2.5,2]) + t4_plane = Rectangle(color = PURPLE, fill_opacity=0.3).scale(0.4).rotate(math.radians(180),LEFT).shift([0.9,2.35,0.4]) + + #------------------------------------------- + self.set_camera_orientation(phi = 50 * DEGREES, theta = 45 * DEGREES) + self.wait(1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(s)) + self.wait(1) + self.play(Write(d1)) + self.add_fixed_in_frame_mobjects(d1_text) + self.play(ShowCreation(t_plane)) + self.add_fixed_in_frame_mobjects(t_text) + self.wait(1) + self.play(FadeOut(t_text),Write(d2)) + self.add_fixed_in_frame_mobjects(d2_text) + self.wait(1) + self.play(Write(l1),Write(l2)) + self.play(Write(d2_copy),Write(d1_copy)) + self.wait(1) + self.play(Write(a1),Write(a_x),Write(a_y)) + self.wait(1) + self.play(Write(a2),Write(a3)) + self.wait(1) + self.play(ReplacementTransform(t_plane,t2_plane)) + self.wait(1) + self.play(ReplacementTransform(t2_plane,t3_plane)) + self.wait(1) + self.play(ReplacementTransform(t3_plane,t4_plane)) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file3_Non_Differentiable_Function.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file3_Non_Differentiable_Function.gif Binary files differnew file mode 100644 index 0000000..7581a33 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file3_Non_Differentiable_Function.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file3_Non_Differentiable_Function.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file3_Non_Differentiable_Function.py new file mode 100644 index 0000000..79d0948 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file3_Non_Differentiable_Function.py @@ -0,0 +1,47 @@ +from manimlib.imports import* +import math + +#---- tangent plane does not exists for f(x,y): sqrt(x**2+y**2) at origin + +class TangenttoSurface(ThreeDScene): + def construct(self): + axes = ThreeDAxes().rotate(2.3) + axes2 = ThreeDAxes().scale(2).rotate(2.3).shift([0,0,1.3]) + + #----f(x,y): sqrt(x**2+y**2) + p = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -math.sqrt(u**2+v**2) + ]),v_min = -1,v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [RED_C,TEAL_D], + resolution = (20, 20)).scale(1) + + #----size increased of f(x,y): sqrt(x**2+y**2) + p2 = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -math.sqrt(u**2+v**2) + ]),v_min = -1,v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [RED_C,TEAL_D], + resolution = (20, 20)).scale(3).shift([0,0,0]) + + self.set_camera_orientation(phi = 75 * DEGREES,theta = 40*DEGREES) + + d = Dot([0,0,0],color = '#800000') #---- critical point + d2 = Dot([0,0,1.5],color = '#800000').scale(2) #---- size increased of critical point + + f_text = TextMobject("$f$ is not differentiable at origin,because the surface").scale(0.5).to_corner(UL) + f2_text = TextMobject("is not flat when zoomed in at the origin.").scale(0.5).to_corner(UL).shift(0.5*DOWN) + + self.add(axes) + self.wait(1) + self.play(Write(p),Write(d)) + self.wait(1) + self.move_camera(phi = 50 * DEGREES,theta = 40*DEGREES) + self.wait(1) + self.play(ReplacementTransform(axes,axes2),ReplacementTransform(p,p2),ReplacementTransform(d,d2)) + self.wait(1) + self.add_fixed_in_frame_mobjects(f_text) + self.add_fixed_in_frame_mobjects(f2_text) + self.wait(2) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file4_Tangent plane_at_extrema_and_saddle_point.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file4_Tangent plane_at_extrema_and_saddle_point.gif Binary files differnew file mode 100644 index 0000000..cfe054b --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file4_Tangent plane_at_extrema_and_saddle_point.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file4_Tangent_plane_at_extrema_and_saddle_point.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file4_Tangent_plane_at_extrema_and_saddle_point.py new file mode 100644 index 0000000..d129213 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Tangent-Plane-Approximations/file4_Tangent_plane_at_extrema_and_saddle_point.py @@ -0,0 +1,62 @@ +from manimlib.imports import* + +class TangenttoSurface(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + + #----graph of first function f(x,y) = -x**2-y**2 + f = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2-v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [YELLOW_D, YELLOW_E], + resolution = (20, 20)).scale(1) + f_text = TextMobject("Tangent plane at relative maxima").to_corner(UL).scale(0.5) + + #----graph of second function f(x,y) = -x**2+y**2 + f2 = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2+v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [RED_D, RED_E], + resolution = (20, 20)).scale(1) + f2_text = TextMobject("Tangent plane at saddle point").to_corner(UL).scale(0.5) + + #----graph of third function f(x,y) = x**2+y**2 + f3 = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2+v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [GREEN_D, GREEN_E], + resolution = (20, 20)).scale(1) + f3_text = TextMobject("Tangent plane at relative minima").to_corner(UL).scale(0.5) + + self.set_camera_orientation(phi = 75 * DEGREES, theta = -45 * DEGREES ) + d = Dot(np.array([0,0,0]), color = '#800000') #---- critical point + + r = Rectangle(height = 2,breadth = 1,color = PURPLE).scale(0.5) + + self.begin_ambient_camera_rotation(rate = 0.3) + self.add(axes) + self.play(Write(f),Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(f_text) + self.play(ShowCreation(r)) + self.wait(1) + self.play(FadeOut(r),FadeOut(f),FadeOut(d),FadeOut(f_text)) + self.wait(1) + self.play(Write(f2),Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(f2_text) + self.play(ShowCreation(r)) + self.wait(1) + self.play(FadeOut(r),FadeOut(f2),FadeOut(d),FadeOut(f2_text)) + self.wait(1) + self.play(Write(f3),Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(f3_text) + self.play(ShowCreation(r)) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/README.md b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/README.md new file mode 100644 index 0000000..96b32bf --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/README.md @@ -0,0 +1,27 @@ +<h1><div align=”center”><b>SubTopic: The Second Derivative Test</b></h1></div> +<br/></br> + +<tab>file1_Second_order_partial_derivatives + + +<br/></br> +<br/></br> + +<tab>file2_Nondegenerate_Hessian_Matrix + + +<br/></br> +<br/></br> + +<tab>file3_Degenerate_Hessian_Matrix + + +<br/></br> +<br/></br> + +<tab>file4_Contour_Diagram + + +<br/></br> +<br/></br> + diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.gif Binary files differnew file mode 100644 index 0000000..3471e4d --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.py new file mode 100644 index 0000000..84052cc --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.py @@ -0,0 +1,78 @@ +from manimlib.imports import* + +#---- graphs of second-order partial derivatives of a function +class SurfacesAnimation(ThreeDScene): + def construct(self): + + axes = ThreeDAxes() + x_label = TextMobject('$x$').shift([5,0.5,0]) #---- x axis + y_label = TextMobject('$y$').shift([0.5,4,0]).rotate(-4.5) #---- y axis + + #---- surface of function: f(x,y) = (x^2+y^2)^2 + surface_f = ParametricSurface( + lambda u, v: np.array([ + u, + v, + ((u**2)+(v**2))**2 + ]),v_min=-1,v_max=1,u_min=-1,u_max=1,checkerboard_colors=[GREEN_D, GREEN_E]).scale(1) + + #---- surface of second-order partial derivative f_xx + surface_fxx = ParametricSurface( + lambda u, v: np.array([ + u, + v, + (3*u**2)+(v**2) + ]),v_min=-1,v_max=1,u_min=-1,u_max=1,checkerboard_colors=[YELLOW_D, YELLOW_E]).shift([0,0,0]).scale(0.6) + + #---- surface of second-order partial derivative f_yy + surface_fyy = ParametricSurface( + lambda u, v: np.array([ + u, + v, + (u**2)+(3*v**2) + ]),v_min=-1,v_max=1,u_min=-1,u_max=1,checkerboard_colors=[PURPLE_D, PURPLE_E]).scale(0.6).shift([0,0,0]) + + #---- surface of second-order partial derivative f_xy = f_yx + surface_fxy = ParametricSurface( + lambda u, v: np.array([ + u, + v, + 8*u*v + ]),v_min=-1,v_max=1,u_min=-1,u_max=1,checkerboard_colors=[TEAL_D, TEAL_E]).scale(0.6) + + f_text= TextMobject("$f(x,y) = (x^2+y^2)^2$",color = GREEN).scale(0.7).to_corner(UL) + + fxx_text= TextMobject("$f_{xx} = 12x^2+4y^2$ (Concavity along x axis)",color = YELLOW).scale(0.5).to_corner(UL) + + fyy_text= TextMobject("$f_{yy} = 4x^2+12y^2$(Concavity along y axis)",color = PURPLE).scale(0.5).to_corner(UL) + + fxy_text= TextMobject("$f_{xy} = f_{yx} = 8xy$ (Twisting of the function)",color = TEAL).scale(0.5).to_corner(UL) + + + self.set_camera_orientation(phi = 40 * DEGREES, theta = 45 * DEGREES) + self.begin_ambient_camera_rotation(rate = 0.1) + self.add_fixed_in_frame_mobjects(f_text) + self.add(axes) + self.add(x_label) + self.add(y_label) + self.wait(1) + self.play(Write(surface_f)) + self.wait(2) + self.play(FadeOut(f_text)) + + + self.play(ReplacementTransform(surface_f,surface_fxx)) + + self.add_fixed_in_frame_mobjects(fxx_text) + self.wait(2) + self.play(FadeOut(fxx_text)) + + self.play(ReplacementTransform(surface_fxx,surface_fyy)) + self.add_fixed_in_frame_mobjects(fyy_text) + self.wait(2) + self.play(FadeOut(fyy_text)) + + self.play(ReplacementTransform(surface_fyy,surface_fxy)) + self.move_camera(phi = 35 * DEGREES, theta = 80 * DEGREES) + self.add_fixed_in_frame_mobjects(fxy_text) + self.wait(2) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Nondegenerate_Hessian_Matrix.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Nondegenerate_Hessian_Matrix.gif Binary files differnew file mode 100644 index 0000000..0d58b4f --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Nondegenerate_Hessian_Matrix.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Nondegenerate_Hessian_Matrix.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Nondegenerate_Hessian_Matrix.py new file mode 100644 index 0000000..32c1559 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Nondegenerate_Hessian_Matrix.py @@ -0,0 +1,158 @@ +from manimlib.imports import* +import math as m + +class Minima(ThreeDScene): + def construct(self): + + heading = TextMobject("Nondegenerate Hessian Matrix",color = BLUE) + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis + + h_text = TextMobject("Case 1: $\\frac{\\partial^2 f}{\\partial x^2}>0$ and $\\frac{\\partial^2 f}{\\partial y^2}>0$").scale(1) + + #---- determiniant of Hessian Matrix + hessian_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -0.5*m.exp(-u**2-v**2) + ]),u_min = -PI, u_max = PI, v_min = -PI, v_max =PI).set_color(TEAL).shift([0,0,0]).scale(1).fade(0.2) + + det_text= TextMobject("$det \\hspace{1mm} H = (\\frac{\\partial^2 f}{\\partial x^2})(\\frac{\\partial^2 f}{\\partial y^2})-(\\frac{\\partial^2 f}{\\partial x \\partial y})^2 $").to_corner(UL).scale(0.7) + + #---- function f(x,y) + f_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2+v**2 + ]),u_min = -1.3, u_max = 1.3, v_min = -1.3, v_max = 1.3).set_color(TEAL).shift([0,0,-0.5]) + + f_text= TextMobject("surface of the function").to_corner(UL).scale(0.8) + + d = Dot(color = "#800000").shift([0,0,-0.52]) #---- critical point + + self.set_camera_orientation(phi = 75*DEGREES, theta = 40*DEGREES) + self.add_fixed_in_frame_mobjects(heading) + self.wait(1) + self.play(FadeOut(heading)) + self.wait(1) + self.add_fixed_in_frame_mobjects(h_text) + self.wait(1) + self.play(FadeOut(h_text)) + self.wait(1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(hessian_surface)) + self.wait(1) + self.add_fixed_in_frame_mobjects(det_text) + self.move_camera(phi = 90*DEGREES, theta= 60*DEGREES) + self.play(Write(d)) + self.wait(1) + self.play(FadeOut(det_text),ReplacementTransform(hessian_surface,f_surface)) + self.wait(1) + self.add_fixed_in_frame_mobjects(f_text) + self.wait(1) + self.play(FadeOut(f_text),FadeOut(f_surface),FadeOut(axes),FadeOut(label_x),FadeOut(label_y),FadeOut(d)) + +class Maxima(ThreeDScene): + def construct(self): + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis + + h_text = TextMobject("Case 2: $\\frac{\\partial^2 f}{\\partial x^2}<0$ and $\\frac{\\partial^2 f}{\\partial y^2}<0$").scale(1) + + #---- determiniant of Hessian Matrix + hessian_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + 0.5*m.exp(-u**2-v**2) + ]),u_min = -PI, u_max = PI, v_min = -PI, v_max =PI).set_color(TEAL).shift([0,0,0]).scale(1).fade(0.2) + + det_text= TextMobject("$det \\hspace{1mm} H = (\\frac{\\partial^2 f}{\\partial x^2})(\\frac{\\partial^2 f}{\\partial y^2})-(\\frac{\\partial^2 f}{\\partial x \\partial y})^2 $").to_corner(UL).scale(0.7) + + #---- function g(x,y) + g_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2-v**2 + ]),u_min = -1.3, u_max = 1.3, v_min = -1.3, v_max = 1.3).set_color(TEAL).shift([0,0,0.5]) + + g_text= TextMobject("surface of the function").to_corner(UL).scale(0.8) + + d = Dot(color = "#800000").shift([0,0,0.5]) #---- critical point + + self.set_camera_orientation(phi = 75*DEGREES, theta = 40*DEGREES) + self.add_fixed_in_frame_mobjects(h_text) + self.wait(1) + self.play(FadeOut(h_text)) + self.wait(1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(hessian_surface)) + self.wait(1) + self.add_fixed_in_frame_mobjects(det_text) + self.play(Write(d)) + self.wait(1) + self.play(FadeOut(det_text),ReplacementTransform(hessian_surface,g_surface)) + self.wait(1) + self.add_fixed_in_frame_mobjects(g_text) + self.wait(1) + self.play(FadeOut(g_text),FadeOut(g_surface),FadeOut(axes),FadeOut(label_x),FadeOut(label_y),FadeOut(d)) + +class SaddlePoint(ThreeDScene): + def construct(self): + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis + + h_text = TextMobject("Case 3: $\\frac{\\partial^2 f}{\\partial x^2}$ and $\\frac{\\partial^2 f}{\\partial y^2}$ have opposite signs").scale(1) + + #---- determiniant of Hessian Matrix + hessian_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + m.exp(0.5*u**2-0.5*v**2) + ]),u_min = -1.2, u_max = 1.2, v_min = -2.5, v_max = 2.5).set_color(TEAL).shift([0,0,-1]).scale(1).fade(0.2) + + det_text= TextMobject("$det \\hspace{1mm} H = (\\frac{\\partial^2 f}{\\partial x^2})(\\frac{\\partial^2 f}{\\partial y^2})-(\\frac{\\partial^2 f}{\\partial x \\partial y})^2 $").to_corner(UL).scale(0.7) + + #---- function p(x,y) + p_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2-v**2 + ]),u_min = -1, u_max = 1, v_min = -1, v_max =1).set_color(TEAL).shift([0,0,0]).scale(2) + + p_text= TextMobject("surface of the function").to_corner(UL).scale(0.8) + + d = Dot(color = "#800000").shift([0,0,0]) #---- critical point + + self.set_camera_orientation(phi = 80*DEGREES, theta = 60*DEGREES) + self.add_fixed_in_frame_mobjects(h_text) + self.wait(1) + self.play(FadeOut(h_text)) + self.wait(1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.wait(1) + self.play(Write(hessian_surface)) + self.play(Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(det_text) + self.wait(2) + self.play(FadeOut(det_text),ReplacementTransform(hessian_surface,p_surface)) + self.add_fixed_in_frame_mobjects(p_text) + self.wait(2) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.gif Binary files differnew file mode 100644 index 0000000..5aae300 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py new file mode 100644 index 0000000..9310553 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py @@ -0,0 +1,45 @@ +from manimlib.imports import* +import math as m + +class DegenerateHessian(ThreeDScene): + def construct(self): + + heading = TextMobject("Degenerate Hessian Matrix",color = BLUE) + + h_text = TextMobject("For $det \\hspace{1mm} H = 0$, the surface of the function at the critical point would be flat.").scale(0.7) + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis + + #---- function f(x,y) + f_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**4-v**4 + ]),u_min = -0.8, u_max = 0.8, v_min = -0.8, v_max = 0.8).set_color(TEAL).shift([0,0,-0.5]).scale(2) + + f_text= TextMobject("surface of the function").to_corner(UL).scale(0.5) + + d = Dot(color = "#800000").shift([0,0,-0.5]) #---- critical point + plane = Square(color = YELLOW,fill_opacity= 0.2).shift([0,0,-0.5]).scale(1.3) + + self.set_camera_orientation(phi = 70*DEGREES, theta = 45*DEGREES) + self.add_fixed_in_frame_mobjects(heading) + self.wait(1) + self.play(FadeOut(heading)) + self.add_fixed_in_frame_mobjects(h_text) + self.wait(2) + self.play(FadeOut(h_text)) + self.wait(1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(f_surface)) + self.add_fixed_in_frame_mobjects(f_text) + self.wait(1) + self.play(Write(d)) + self.wait(1) + self.play(Write(plane)) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.gif Binary files differnew file mode 100644 index 0000000..41068e2 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.py new file mode 100644 index 0000000..d3084e2 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.py @@ -0,0 +1,120 @@ +from manimlib.imports import* + +#---- contour diagram animation +class ContourDiagram(ThreeDScene): + def construct(self): + + heading = TextMobject("CONTOUR DIAGRAM", color = YELLOW).scale(1) + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #---- surface of a paraboloid + surface = ParametricSurface( + lambda u, v: np.array([ + np.cos(v)*u, + np.sin(v)*u, + u**2 + ]),v_min = -2, v_max = 2, u_min = -2, u_max = 2, checkerboard_colors = [GREEN_B,GREEN_C,GREEN_D,GREEN_E]).shift([0,0,0]).scale(0.5) + + #---- first contour projection + contour1 = ParametricSurface( + lambda u, v: np.array([ + np.cos(TAU * v), + np.sin(TAU * v), + 2*(1 - 2.5*u) + ])).fade(0.5).scale(0.21).shift([0,0,1.01]) + + #---- first contour line + c_1 = Circle(color = BLUE).scale(0.21).shift([0,0,0]).rotate(0.1,DOWN) + + #------------------------------------------------- + + #---- second contour projection + contour2 = ParametricSurface( + lambda u, v: np.array([ + np.cos(TAU * v), + np.sin(TAU * v), + 2*(1 - 1.6*u) + ])).fade(0.5).scale(0.41).shift([0,0,0.3]).set_color(RED) + + #---- second contour line + c_2 = Circle(color = RED).scale(0.41).shift([0,0,0]).rotate(0.1,DOWN) + + #------------------------------------------------- + + #---- third contour projection + contour3 = ParametricSurface( + lambda u, v: np.array([ + np.cos(TAU * v), + np.sin(TAU * v), + 2*(1 - 1.5*u) + ])).fade(0.5).scale(0.61).shift([0,0,0.4]).set_color(YELLOW) + + #---- third contour line + c_3 = Circle(color = YELLOW).scale(0.61).shift([0,0,0]) + + #------------------------------------------------- + + #---- fourth contour projection + contour4 = ParametricSurface( + lambda u, v: np.array([ + np.cos(TAU * v), + np.sin(TAU * v), + 2*(1 - 1.5*u) + ])).fade(0.7).scale(0.81).shift([0,0,0.7]).set_color(PINK) + + #---- fourth contour line + c_4 = Circle(color = PINK).scale(0.81).shift([0,0,0]) + + #------------------------------------------------- + + #---- fifth contour projection + contour5 = ParametricSurface( + lambda u, v: np.array([ + np.cos(TAU * v), + np.sin(TAU * v), + 2*(1 - 1.5*u) + ])).fade(0.7).scale(1.01).shift([0,0,1]).set_color(PURPLE) + + #---- fifth contour line + c_5 = Circle(color = PURPLE).scale(1.01).shift([0,0,0]) + + c_text= TextMobject("Contour Lines").scale(0.5).shift(2*DOWN) + s = Square().scale(1.3) + + self.set_camera_orientation(phi = 75 * DEGREES, theta = 10 * DEGREES) + self.add_fixed_in_frame_mobjects(heading) + self.wait(1) + self.play(FadeOut(heading)) + self.wait(1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(surface)) + self.wait(1) + self.add(contour1) + self.wait(1) + self.play(Write(c_1)) + self.play(ReplacementTransform(contour1,contour2)) + self.wait(1) + self.play(Write(c_2)) + self.play(ReplacementTransform(contour2,contour3)) + self.wait(1) + self.play(Write(c_3)) + self.play(ReplacementTransform(contour3,contour4)) + self.wait(1) + self.play(Write(c_4)) + self.play(ReplacementTransform(contour4,contour5)) + self.wait(1) + self.play(Write(c_5)) + self.wait(1) + self.play(FadeOut(contour5),FadeOut(axes),FadeOut(label_x),FadeOut(label_y),FadeOut(surface),FadeOut(contour5),FadeOut(contour4),FadeOut(contour3),FadeOut(contour2),FadeOut(contour1)) + self.wait(1) + self.move_camera(phi=0 * DEGREES,theta= 90*DEGREES) + self.wait(1) + self.add_fixed_in_frame_mobjects(c_text) + self.wait(1) + self.play(ShowCreation(s),FadeOut(c_text)) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/README.md b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/README.md new file mode 100644 index 0000000..ce4da11 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/README.md @@ -0,0 +1,34 @@ +<h1><div align=”center”><b>SubTopic: Total Differential</b></h1></div> +<br/></br> + +<tab>file1_Visualization_of_dz + + +<br/></br> +<br/></br> + +<tab>file2_Differentials + + + +<br/></br> +<br/></br> + +<tab>file3_Total_differential_of_z + + +<br/></br> +<br/></br> + +<tab>file4_total_differential_change + + +<br/></br> +<br/></br> + +<tab>file5_Total_differential_approximation + +  + +<br/></br> +<br/></br> diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file1_Visualization_of_dz.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file1_Visualization_of_dz.gif Binary files differnew file mode 100644 index 0000000..2e148af --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file1_Visualization_of_dz.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file1_Visualization_of_dz.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file1_Visualization_of_dz.py new file mode 100644 index 0000000..1fdd0b9 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file1_Visualization_of_dz.py @@ -0,0 +1,59 @@ +from manimlib.imports import* + +#---- visualization of total differential dz between two points lying on the surface of the function +class differentialdz(ThreeDScene): + + def construct(self): + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]).fade(0.4) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5).fade(0.4) #---- y axis + + #---- surface of the funtion f(x,y) + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2+v**2 + ]),u_min=-1,u_max=1, v_min=-1,v_max=1).set_color("#FF69B4").fade(0.6).scale(2).shift(3*UP+1*LEFT) + + d = Dot([1.4,1.75,1],color = '#00FFFF').rotate(1.571,UP) #---- point on the surface + d2 = Dot([2,2,1],color = '#00FFFF').rotate(1.571,UP) #---- point on the surface + + p1 = TextMobject("$P_1$",color ='#ADFF2F').scale(0.6).shift(2*RIGHT+1*UP) + p2 = TextMobject("$P_2$",color = '#ADFF2F').scale(0.6).shift(2.6*RIGHT+0.9*UP) + + l = DashedLine(color = '#800000').rotate(1.571,UP).scale(1).shift(1.7*UP+1.6*RIGHT) + l2 = DashedLine(color = '#800000').rotate(1.571,UP).scale(0.8).shift(2.26*UP+1.2*RIGHT) + + l_text = TextMobject("$(x_1,y_1)$",color = '#ADFF2F').scale(0.6).shift(2*RIGHT+1.6*DOWN) + l2_text = TextMobject("$(x_2,y_2)$",color = '#ADFF2F').scale(0.6).shift(2.7*RIGHT+1.2*DOWN) + + a = Arrow(color = '#FFFACD').scale(0.7).rotate(1.38,RIGHT).shift(2.5*LEFT+3.1*UP) + + a_text = TextMobject("$dz$",color='#800000').scale(0.5).shift(2.3*RIGHT+0.5*UP) + + plane = Rectangle(color = '#E6E6FA',fill_opacity = 1).scale(3).shift(1*RIGHT+3*UP).fade(0.9) + + label = TextMobject("$z = f(x,y)$").scale(0.6).shift(3.5*RIGHT+1.8*UP) + + self.set_camera_orientation(phi=75*DEGREES,theta=-10*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.wait(1) + self.play(Write(plane)) + self.play(Write(surface)) + self.add_fixed_in_frame_mobjects(label) + self.wait(1) + self.play(ShowCreation(l),ShowCreation(l2),Write(d),Write(d2)) + self.wait(1) + self.add_fixed_in_frame_mobjects(p1) + self.add_fixed_in_frame_mobjects(p2) + self.wait(1) + self.add_fixed_in_frame_mobjects(l_text) + self.add_fixed_in_frame_mobjects(l2_text) + self.play(ShowCreation(a)) + self.wait(1) + self.add_fixed_in_frame_mobjects(a_text) + self.wait(2) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file2_Differentials.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file2_Differentials.gif Binary files differnew file mode 100644 index 0000000..6baf271 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file2_Differentials.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file2_Differentials.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file2_Differentials.py new file mode 100644 index 0000000..1025210 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file2_Differentials.py @@ -0,0 +1,77 @@ +from manimlib.imports import* + +#---- visualization of the differentials along the axes +class differentials(ThreeDScene): + def construct(self): + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.3,0]).fade(0.4) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5).fade(0.4) #---- y axis + + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2+v**2 + ]),u_min=-1,u_max=1, v_min=-1,v_max=1).set_color("#FF69B4").shift([0,2.5,0.3]).scale(1.2) #----surface z = f(x,y) + + + + plane = Rectangle(color = '#E6E6FA',fill_opacity = 1).scale(3).shift(-1*RIGHT+3*UP).fade(0.9) + + d = Dot([1,2,1],color = '#9400D3').rotate(1.571,UP) + d2 = Dot([2,2.9,1],color = '#9400D3').rotate(1.571,UP) + + p1 = TextMobject("$P_1$",color ='#ADFF2F').scale(0.6).shift(2*RIGHT+1*UP) + p2 = TextMobject("$P_2$",color = '#ADFF2F').scale(0.6).shift(2.6*RIGHT+0.4*UP) + + + l1 = DashedLine(color = '#00BFFF').scale(1.6).shift(3.5*UP+3.25*LEFT).rotate(1.571) + l2 = DashedLine(color = '#00BFFF').scale(1).shift(4*UP+2*LEFT).rotate(1.571) + + label_dz= TextMobject("$dz$").scale(0.4).shift(5.3*RIGHT+0.4*UP) + + + l3 = Line(color = '#FFDAB9').scale(0.8).shift(1.95*UP+0.7*RIGHT).rotate(1.571,DOWN).fade(0.2) + l4 = Line(color = '#FFDAB9').scale(0.6).shift(2.86*UP+0.9*RIGHT).rotate(1.571,DOWN).fade(0.2) + + line_y1 = DashedLine(color = '#00BFFF').scale(1.3).shift(0.82*UP+3.25*RIGHT).rotate(1.571) + line_y2 = DashedLine(color = '#00BFFF').scale(1.7).shift(1.2*UP+2.8*RIGHT).rotate(1.571) + + label_dy= TextMobject("$dy$").scale(0.6).shift(3*RIGHT+0.8*DOWN).rotate(math.radians(90)) + + line_x1 = DashedLine(color = '#00BFFF').scale(1.5).shift(2.2*UP+1.6*RIGHT).rotate(1.571,RIGHT) + line_x2 = DashedLine(color = '#00BFFF').scale(1.2).shift(2.9*UP+1.6*RIGHT).rotate(1.571,RIGHT) + + label_dx= TextMobject("$dx$").scale(0.4).shift(-0.4*UP+2.5*RIGHT) + + label = TextMobject("$f(x,y)$").scale(0.6).shift(4*RIGHT+3*UP) + + + self.set_camera_orientation(phi=75*DEGREES,theta=10*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(plane)) + self.play(Write(surface)) + self.add_fixed_in_frame_mobjects(label) + self.wait(1) + self.play(Write(d),Write(d2)) + self.add_fixed_in_frame_mobjects(p1) + self.add_fixed_in_frame_mobjects(p2) + self.wait(1) + self.play(Write(l1)) + self.play(Write(l2)) + self.add_fixed_in_frame_mobjects(label_dz) + self.wait(1) + self.play(Write(l3)) + self.play(Write(l4)) + self.wait(1) + self.play(Write(line_y1)) + self.play(Write(line_y2)) + self.play(ShowCreation(label_dy)) + self.wait(1) + self.play(Write(line_x1)) + self.play(Write(line_x2)) + self.add_fixed_in_frame_mobjects(label_dx) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file3_Total_differential_of_z.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file3_Total_differential_of_z.gif Binary files differnew file mode 100644 index 0000000..a54d2da --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file3_Total_differential_of_z.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file3_Total_differential_of_z.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file3_Total_differential_of_z.py new file mode 100644 index 0000000..b8d6f96 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file3_Total_differential_of_z.py @@ -0,0 +1,100 @@ +from manimlib.imports import* + +#---- visualization of total differential definition +class totaldifferential(ThreeDScene): + def construct(self): + axes = ThreeDAxes().fade(0.5) + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2+v**2 + ]),u_min=-1,u_max=1, v_min=-1,v_max=1).set_color("#FF69B4").fade(0.6).shift([1,0.8,1.5]).scale(2) + + plane = Rectangle(color = '#E6E6FA',fill_opacity = 1).scale(3).shift(-1*RIGHT+3*UP).fade(0.9) + label_x = TextMobject("$x$").shift(5*RIGHT+0.4*DOWN).rotate(1.571) + label_y = TextMobject("$y$").shift(0.3*DOWN+5.6*RIGHT).scale(0.5) + label_z = TextMobject("$z$").shift(3.5*UP+0.2*LEFT).scale(0.5) + + s1 = Square(color = '#00FF00',fill_opacity=0.4).shift([1,1,0]) + s2 = Square(color = '#00FF00',fill_opacity=0.4).shift([1,1,3]).scale(0.95) + + l1 = Line([2,0,3],[2,0,0],color = '#FFFACD') + l2 = Line([0,2,3],[0,2,0],color = '#FFFACD') + l3 = Line([2,1.95,3],[2,2,0],color = '#FFFACD') + + d1 = Dot([2,0,1.5],color = '#FFD700').rotate(1.571,UP) + d1_text = TextMobject("$P1$").scale(0.4).shift(1.2*LEFT+1.1*UP) + + d2 = Dot([0,2,3],color = '#FFD700').rotate(1.571,UP) + d2_text = TextMobject("$P2$").scale(0.4).shift(2.3*RIGHT+3.1*UP) + + d3 = Dot([2,2,2],color = '#FFD700').rotate(1.571,UP) + d3_text = TextMobject("$Q$").scale(0.4).shift([1.6,-1,0]+2.5*UP) + + s3 = Square().shift([1,1,1.5]).scale(0.95) + s4 = Square().shift([1,1,2]).scale(0.95) + + m1_line = DashedLine([2,0,1.5],[2,2,2],color = '#87CEEB') + m2_line = DashedLine([2,2,2],[0,2,3],color = '#87CEEB') + + dx_line = Line([2,2,0],[4,2,0],color = '#00FF7F') + dy_line = Line([2,2,0],[2,4,0],color = '#00FF7F') + + dx = DashedLine([3.5,0,0],[3.5,2,0],color = '#87CEEB') + dy = DashedLine([0,3.5,0],[2,3.5,0],color = '#87CEEB') + + dx_text = TextMobject("$dx$").scale(0.8).shift([4,1,0]).rotate(1.571) + dy_text = TextMobject("$dy$").scale(0.8).shift([1,3.8,0]).rotate(math.radians(180)) + + parx_line = Line([0,2,1.5],[0,5,1.5],color = '#00FF7F') + parm_line = Line([0,2,2],[0,5,2],color = '#00FF7F') + pary_line = Line([0,2.1,3],[0,5,3],color = '#00FF7F') + + delx = DashedLine([0,4,2],[0,4,1.5],color = '#F0F8FF') + dely = DashedLine([0,4,3],[0,4,2],color = '#FAEBD7') + + dely_text = TextMobject("$\\frac{\\partial z}{\\partial y}dy$").shift(4.6*RIGHT+2.3*UP).scale(0.4) + delx_text = TextMobject("$\\frac{\\partial z}{\\partial x}dx$").shift(4.6*RIGHT+1.4*UP).scale(0.4) + + + self.set_camera_orientation(phi=75*DEGREES,theta=20*DEGREES) + self.add(axes) + self.play(Write(plane)) + self.play(ShowCreation(label_x)) + self.add_fixed_in_frame_mobjects(label_y) + self.add_fixed_in_frame_mobjects(label_z) + self.wait(1) + self.play(Write(surface)) + self.play(ShowCreation(d1)) + self.add_fixed_in_frame_mobjects(d1_text) + self.play(ShowCreation(d2)) + self.add_fixed_in_frame_mobjects(d2_text) + self.wait(1) + self.play(Write(s2)) + self.wait(1) + self.play(Write(l1),Write(l2),Write(l3)) + self.wait(1) + self.play(Write(s1)) + self.wait(1) + self.play(FadeOut(surface)) + self.play(ShowCreation(d3)) + self.add_fixed_in_frame_mobjects(d3_text) + self.play(ShowCreation(m1_line)) + self.play(ShowCreation(m2_line)) + self.wait(1) + self.play(ShowCreation(dx_line),ShowCreation(dx),ShowCreation(dx_text)) + self.wait(1) + self.play(ShowCreation(dy_line),ShowCreation(dy),ShowCreation(dy_text)) + self.wait(2) + self.play(Write(s3)) + self.play(Write(s4)) + self.wait(1) + self.play(ShowCreation(parx_line),ShowCreation(parm_line),ShowCreation(pary_line)) + self.wait(1) + self.play(ShowCreation(dely)) + self.add_fixed_in_frame_mobjects(dely_text) + self.wait(1) + self.play(ShowCreation(delx)) + self.add_fixed_in_frame_mobjects(delx_text) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file4_total_differential_change.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file4_total_differential_change.gif Binary files differnew file mode 100644 index 0000000..f2227a8 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file4_total_differential_change.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file4_total_differential_change.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file4_total_differential_change.py new file mode 100644 index 0000000..78e41a2 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file4_total_differential_change.py @@ -0,0 +1,54 @@ +from manimlib.imports import* + + +class firstScene(ThreeDScene): + + def construct(self): + + axes = ThreeDAxes() + + s = Rectangle(color = '#F08080',fill_opacity=1).fade(0.7).shift(1.9*UP+5*LEFT).scale(0.9)#----surface z = f(x,y) + + s2= Rectangle(color = '#F08080',fill_opacity=1).fade(0.7).shift(2.4*UP+3.1*RIGHT).scale(0.6) #----reflection of the surface on the x-y plane + + l1 = DashedLine(color = '#AFEEEE').rotate(1.571,UP).scale(1).shift(1.53*UP+1.5*RIGHT) + l2 = DashedLine(color = '#AFEEEE').rotate(1.571,UP).scale(1).shift(2.9*UP+1.4*RIGHT) + l3 = DashedLine(color = '#AFEEEE').rotate(1.571,UP).scale(1).shift(1.5*UP-1.6*RIGHT) + l4 = DashedLine(color = '#AFEEEE').rotate(1.571,UP).scale(1).shift(2.9*UP-1.75*RIGHT) + + + l1_text = TextMobject("$(x+\\triangle x,y)$").shift(RIGHT+1.7*DOWN).scale(0.4) + l2_text = TextMobject("$(x+\\triangle x,y+\\triangle y)$").shift(3*RIGHT+1.8*DOWN).scale(0.4) + l3_text = TextMobject("$f(x,y)$").shift(1.6*RIGHT+1.5*UP).scale(0.4) + l4_text = TextMobject("$(x,y+\\triangle y)$").shift(3.5*RIGHT+0.7*DOWN).scale(0.4) + + label_x = TextMobject("$x$").shift(5*RIGHT+0.4*DOWN) + label_y = TextMobject("$y$").shift(5*UP-0.6*RIGHT) + + self.add(axes) + self.set_camera_orientation(phi=75*DEGREES,theta=10*DEGREES) + self.wait(1) + self.play(ShowCreation(label_x),ShowCreation(label_y)) + self.play(Write(s)) + self.wait(1) + self.add_fixed_in_frame_mobjects(l3_text) + self.wait(1) + self.play(Write(l3)) + self.wait(1) + self.play(Write(l1)) + self.add_fixed_in_frame_mobjects(l1_text) + self.wait(1) + self.play(Write(l2)) + self.add_fixed_in_frame_mobjects(l2_text) + self.wait(1) + self.play(Write(l4)) + self.add_fixed_in_frame_mobjects(l4_text) + self.wait(1) + self.play(Write(s2)) + self.wait(1) + + + + + + diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file5_Total_differential_approximation.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file5_Total_differential_approximation.gif Binary files differnew file mode 100644 index 0000000..ebbf240 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file5_Total_differential_approximation.gif diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file5_Total_differential_approximation.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file5_Total_differential_approximation.py new file mode 100644 index 0000000..e7b39bb --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Total-Differential/file5_Total_differential_approximation.py @@ -0,0 +1,52 @@ +from manimlib.imports import* + +#---- approximation value of function between two points using total differentials +class approximation(ThreeDScene): + + def construct(self): + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.3,0]).fade(0.4) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5).fade(0.4) #---- y axis + + surface = ParametricSurface( + lambda u, v: np.array([ + np.sin(u), + v, + -u**2-v + ]),u_min=-1,u_max=1, v_min=-1,v_max=1).set_color("#00008B").scale(2).shift(3.8*UP+2*LEFT) + + d = Dot([1.4,1.75,1],color = '#00FFFF').rotate(1.571,UP) + d2 = Dot([2,2,1],color = '#00FFFF').rotate(1.571,UP) + + l = DashedLine(color = '#800000').rotate(1.571,UP).scale(1).shift(1.7*UP+1.6*RIGHT) + l2 = DashedLine(color = '#800000').rotate(1.571,UP).scale(0.8).shift(2.26*UP+1.2*RIGHT) + + l_text = TextMobject("$(x_1,y_1)$",color = '#ADFF2F').scale(0.6).shift(2*RIGHT+1.6*DOWN) + l2_text = TextMobject("$(x_2,y_2)$",color = '#ADFF2F').scale(0.6).shift(2.7*RIGHT+1.2*DOWN) + + plane = Rectangle(color = '#E6E6FA',fill_opacity = 1).scale(3).shift(1*RIGHT+3*UP).fade(0.9) + + tangentplane = Rectangle(color = '#E6E6FA',fill_opacity = 1).scale(1.1).shift(2*LEFT+3.4*UP).fade(0.5).rotate(0.8,RIGHT) + tangentplane_text = TextMobject("Tangent Plane").scale(0.4).shift(3*RIGHT+1*UP) + + label = TextMobject("$z = f(x,y)$").scale(0.6).shift(4*RIGHT+3*UP) + + self.set_camera_orientation(phi=75*DEGREES,theta=-10*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.wait(1) + self.play(Write(plane)) + self.wait(1) + self.play(Write(surface)) + self.add_fixed_in_frame_mobjects(label) + self.wait(1.5) + self.play(ShowCreation(l),ShowCreation(l2),Write(d),Write(d2)) + self.wait(1) + self.add_fixed_in_frame_mobjects(l_text) + self.add_fixed_in_frame_mobjects(l2_text) + self.wait(1) + self.play(Write(tangentplane)) + self.add_fixed_in_frame_mobjects(tangentplane_text) + self.wait(2) diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_examples.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_examples.gif Binary files differdeleted file mode 100644 index 11f66f1..0000000 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_examples.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_plot_sphere.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_plot_sphere.gif Binary files differdeleted file mode 100644 index ad7582c..0000000 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_plot_sphere.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_respresntation.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_respresntation.gif Binary files differdeleted file mode 100644 index a173bda..0000000 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_respresntation.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_examples.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_examples.py deleted file mode 100644 index 7322e47..0000000 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_examples.py +++ /dev/null @@ -1,69 +0,0 @@ -from manimlib.imports import *
-
-class Examples(GraphScene):
- def construct(self):
-
- rectangle = Rectangle(height = 3, width = 4, color = GREEN)
- square = Square(side_length = 5, color = PURPLE)
- circle = Circle(radius = 2, color = PINK)
- radius = Line(ORIGIN,2*RIGHT)
-
- radius.set_color(RED)
-
- rectangle_area_func = TextMobject(r"$Area = f(Length, Breadth)$")
- rectangle_area_func.scale(0.6)
- square_area_func = TextMobject(r"$Area = f(Length)$")
- circle_area_func = TextMobject(r"$Area = f(r)$")
-
-
- rectangle_area = TextMobject(r"$Area = Length \times Breadth$")
- rectangle_area.scale(0.6)
- square_area = TextMobject(r"$Area = Length^2$")
- circle_area = TextMobject(r"$Area = \pi r^2$")
-
- braces_rect1 = Brace(rectangle, LEFT)
- eq_text1 = braces_rect1.get_text("Length")
- braces_rect2 = Brace(rectangle, UP)
- eq_text2 = braces_rect2.get_text("Breadth")
-
- braces_square = Brace(square, LEFT)
- braces_square_text = braces_square.get_text("Length")
-
- radius_text = TextMobject("r")
- radius_text.next_to(radius,UP)
-
-
-
- self.play(ShowCreation(rectangle))
- self.wait(1)
- self.play(GrowFromCenter(braces_rect1),Write(eq_text1),GrowFromCenter(braces_rect2),Write(eq_text2))
- self.wait(1)
- self.play(Write(rectangle_area_func))
- self.wait(1)
- self.play(Transform(rectangle_area_func, rectangle_area))
- self.wait(1)
- self.play(FadeOut(braces_rect1),FadeOut(eq_text1),FadeOut(braces_rect2),FadeOut(eq_text2),FadeOut(rectangle_area_func))
-
-
- self.play(Transform(rectangle, square))
- self.wait(1)
- self.play(GrowFromCenter(braces_square),Write(braces_square_text))
- self.wait(1)
- self.play(Write(square_area_func))
- self.wait(1)
- self.play(Transform(square_area_func, square_area))
- self.wait(1)
- self.play(FadeOut(braces_square),FadeOut(braces_square_text),FadeOut(square_area_func))
-
-
- self.play(Transform(rectangle, circle))
- self.wait(1)
- self.play(ShowCreation(radius),Write(radius_text))
- self.wait(1)
- self.play(FadeOut(radius_text),FadeOut(radius))
- self.wait(1)
- self.play(Write(circle_area_func))
- self.wait(1)
- self.play(Transform(circle_area_func, circle_area))
- self.wait(1)
- self.play(FadeOut(circle_area_func))
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_plot_sphere.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_plot_sphere.py deleted file mode 100644 index baf08b1..0000000 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_plot_sphere.py +++ /dev/null @@ -1,42 +0,0 @@ -from manimlib.imports import *
-
-class Sphere(ThreeDScene):
- def construct(self):
- axes = ThreeDAxes() # creates a 3D Axis
-
- sphere = ParametricSurface(
- lambda u, v: np.array([
- np.sin(u)*np.cos(v),
- np.sin(u)*np.sin(v),
- np.cos(u)
- ]),u_min=0,u_max=PI,v_min=0,v_max=2*PI,checkerboard_colors=[RED_D, RED_E],
- resolution=(15, 32)).scale(2)
-
-
-
- #self.set_camera_orientation(phi=0 * DEGREES,theta=270*DEGREES)
-
- text3d = TextMobject(r"$f(x,y) \rightarrow Point(x,y,z)$")
- text3d1 = TextMobject(r"$f(x,y) \rightarrow Point(x,y, 1 - x^2 - y^2)$")
- self.add_fixed_in_frame_mobjects(text3d)
- text3d.scale(0.7)
- text3d1.scale(0.7)
- text3d.to_corner(UL)
- text3d1.to_corner(UL)
- text3d.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
- text3d1.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
- self.play(Write(text3d))
- self.wait(1)
-
- self.play(Transform(text3d,text3d1))
- self.add_fixed_in_frame_mobjects(text3d1)
- self.play(FadeOut(text3d))
-
-
- self.set_camera_orientation(phi=75 * DEGREES)
- self.begin_ambient_camera_rotation(rate=0.3)
-
-
- self.add(axes)
- self.play(Write(sphere))
- self.wait(5)
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_respresntation.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_respresntation.py deleted file mode 100644 index 4bfcf21..0000000 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_respresntation.py +++ /dev/null @@ -1,80 +0,0 @@ -from manimlib.imports import *
-
-class MultivariableFunc(Scene):
- def construct(self):
-
- topic = TextMobject("Multivariable Functions")
- topic.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
- topic.scale(2)
-
- self.play(Write(topic))
- self.wait(1)
- self.play(FadeOut(topic))
-
-
- circle = Circle()
- circle.scale(3)
-
- eqn1 = TextMobject(r"f(x,y) = $x^2y$")
- eqn1.set_color(YELLOW)
-
-
-
- number1 = TextMobject("(2,1)")
- number1.move_to(3*UP+ 3*LEFT)
- number1.scale(1.2)
- number1.set_color(GREEN)
-
- output1 = TextMobject("4")
- output1.scale(1.5)
- output1.set_color(BLUE)
-
- eqn1_1 = TextMobject(r"f(2,1) = $2^2(1)$")
- eqn1_1.set_color(YELLOW)
-
-
- self.play(ShowCreation(circle),Write(eqn1))
- self.wait(1)
- self.play(ApplyMethod(number1.move_to, 0.6*LEFT))
- self.play(FadeOut(number1))
- self.play(Transform(eqn1, eqn1_1))
- self.wait(1)
- self.play(ApplyMethod(output1.move_to, 3*DOWN+4*RIGHT))
- self.wait(1)
- self.play(FadeOut(output1))
-
-
- eqn2 = TextMobject(r"f(x,y,z) = $x^2y+2yz$")
- eqn2.set_color(YELLOW)
-
- number2 = TextMobject("(2,1,3)")
- number2.move_to(3*UP+ 3*LEFT)
- number2.scale(1.2)
- number2.set_color(GREEN)
-
- output2 = TextMobject("8")
- output2.scale(1.5)
- output2.set_color(BLUE)
-
- eqn2_1 = TextMobject(r"f(2,1,3) = $2^2(1) + 2(1)(3)$")
- eqn2_1.set_color(YELLOW)
-
- eqn2_2 = TextMobject(r"f(2,1,3) = $2 + 6$")
- eqn2_2.set_color(YELLOW)
-
-
-
- self.play(FadeOut(eqn1))
- self.play(Write(eqn2))
-
- self.wait(1)
- self.play(ApplyMethod(number2.move_to, 1.2*LEFT))
- self.play(FadeOut(number2))
- self.play(Transform(eqn2, eqn2_1))
- self.wait(1)
- self.play(Transform(eqn2, eqn2_2))
- self.wait(1)
- self.play(ApplyMethod(output2.move_to, 3*DOWN+4*RIGHT))
- self.wait(1)
- self.play(FadeOut(output2),FadeOut(eqn2),FadeOut(circle))
- self.wait(2)
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/README.md b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/README.md index b50200d..97a0fb7 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/README.md +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/README.md @@ -4,6 +4,6 @@ Github Account : <a href="https://github.com/nishanpoojary">nishanpoojary</a> ## Sub-Topics Covered:
+ Scalar Functions
+ Multivariable Functions
-+ Limits and continuity of Multivariable Function
++ Multivariable Limits and Continuity
+ Partial Derivatives
+ Directonal Derivatives
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/domain_range.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/domain_range.gif Binary files differdeleted file mode 100644 index d0351e5..0000000 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/domain_range.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_application.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_application.gif Binary files differdeleted file mode 100644 index 831ec8e..0000000 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_application.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_level_curves.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_level_curves.gif Binary files differdeleted file mode 100644 index 2df2fde..0000000 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_level_curves.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_level_curves2.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_level_curves2.gif Binary files differdeleted file mode 100644 index 724c27d..0000000 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_level_curves2.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_parabola_example.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_parabola_example.gif Binary files differdeleted file mode 100644 index 3fdecf4..0000000 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_parabola_example.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/file1_directional_deriv.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/file1_directional_deriv.py new file mode 100644 index 0000000..677d821 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/file1_directional_deriv.py @@ -0,0 +1,85 @@ +from manimlib.imports import *
+
+class GeomRepresen(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ paraboloid = ParametricSurface(
+ lambda u, v: np.array([
+ 3*np.sin(u)*np.cos(v),
+ 3*np.sin(u)*np.sin(v),
+ -0.25*3*3*np.sin(u)*np.sin(u)+2
+ ]),u_min=0,u_max=PI/4,v_min=0,v_max=2*PI, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.7,
+ resolution=(15, 32)).scale(1)
+
+ parabola_curve = ParametricFunction(
+ lambda u : np.array([
+ u,
+ -u,
+ -0.5*(u*u)+2
+ ]),color=PINK,t_min=-1.5,t_max=1.5,
+ )
+
+ circle = Circle(radius = 2.22 , color = BLACK, fill_color = BLUE_C, fill_opacity= 0.3, stroke_width=0.1)
+
+ plane = Polygon(np.array([2.5,-2.5,0]),np.array([-2.5,2.5,0]),np.array([-2.5,2.5,2.5]),np.array([2.5,-2.5,2.5]),np.array([2.5,-2.5,0]), color = BLACK, fill_color = PINK, fill_opacity= 0.2, stroke_width=0.1)
+
+ line = DashedLine(np.array([1,-1,0]), np.array([1,-1,1.5]), color = YELLOW_C)
+
+ tangent_line = Line(np.array([1.5,-1.5,1]), np.array([0.5,-0.5,2]), color = RED_C)
+
+ vector = Arrow(np.array([1,-1,0]), np.array([0.5,-0.5,0]), buff=0.01, color = GREEN_C)
+
+ dot1 =Sphere(radius=0.08).move_to(np.array([1,-1,0])).set_fill(YELLOW_C)
+ dot2 =Sphere(radius=0.08).move_to(np.array([1,-1,1.5])).set_fill(YELLOW_C)
+
+ dot1_lab = TextMobject(r"$P_0$").scale(0.6).move_to(np.array([1,-1,1.8])).set_color(RED_C)
+ dot2_lab = TextMobject(r"$(x_0,y_0)$").scale(0.6).move_to(np.array([1.6,-1,0])).set_color(PURPLE)
+ vector_lab = TextMobject(r"$\hat{u}$").scale(0.8).move_to(np.array([1.2,-0.5,0])).set_color(GREEN_C)
+ domain_lab = TextMobject(r"$D$").scale(0.6).move_to(np.array([1,1,0])).set_color(GREEN_C)
+ func_lab = TextMobject(r"$z = f(x,y)$").scale(0.6).move_to(1*UP + 2.8*RIGHT).set_color(BLUE_C)
+ directional_deriv_lab = TextMobject(r"Slope = $D_{\hat{u}}f(x_0,y_0)$").scale(0.6).move_to(2.2*UP + 1.5*RIGHT).set_color(YELLOW_C)
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.set_camera_orientation(phi=65 * DEGREES, theta = 20*DEGREES)
+
+ self.play(ShowCreation(paraboloid))
+ self.add_fixed_in_frame_mobjects(func_lab)
+ self.wait()
+
+ #self.play(ShowCreation(circle))
+ self.bring_to_front(circle)
+ self.wait()
+ self.add_fixed_orientation_mobjects(domain_lab)
+ self.wait()
+
+ self.play(ShowCreation(plane), ShowCreation(parabola_curve))
+ self.play(ShowCreation(dot1), GrowArrow(line), ShowCreation(dot2))
+ self.add_fixed_orientation_mobjects(dot1_lab)
+ self.wait()
+ self.add_fixed_orientation_mobjects(dot2_lab)
+ self.wait()
+
+ self.play(ShowCreation(tangent_line))
+ self.add_fixed_in_frame_mobjects(directional_deriv_lab)
+ self.wait()
+
+ self.play(GrowArrow(vector))
+ self.add_fixed_orientation_mobjects(vector_lab)
+ self.wait()
+
+
+ self.begin_ambient_camera_rotation(rate=0.1)
+ self.wait(3)
+
+
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/file2_gradient.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/file2_gradient.py new file mode 100644 index 0000000..e9fef50 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/file2_gradient.py @@ -0,0 +1,103 @@ +from manimlib.imports import *
+
+class Gradient(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes() # creates a 3D Axis
+
+
+ quadrant = ParametricSurface(
+ lambda u, v: np.array([
+ 2*np.sin(u)*np.cos(v),
+ 2*np.sin(u)*np.sin(v),
+ 2*np.cos(u)
+ ]),u_min=0,u_max=PI/3,v_min=0,v_max=PI/2,checkerboard_colors=[GREEN_C, GREEN_E],
+ resolution=(15, 32)).scale(1)
+
+ quadrant_curve = ParametricSurface(
+ lambda u, v: np.array([
+ 2*np.sin(u)*np.cos(v),
+ 2*np.sin(u)*np.sin(v),
+ 2*np.cos(u)
+ ]),u_min=34*DEGREES,u_max=38*DEGREES,v_min=0,v_max=PI/2,checkerboard_colors=[YELLOW_C, YELLOW_E],
+ resolution=(15, 32)).scale(1)
+
+
+
+ dot1 =Sphere(radius=0.05).move_to(np.array([1,1,0])).set_fill(YELLOW_C)
+ dot2 =Sphere(radius=0.05).move_to(np.array([1,1,1.732])).set_fill(YELLOW_C)
+
+ dot1_line = DashedLine(np.array([1,1,1.732]), np.array([0,2,2]), color = WHITE)
+ dot1_lab = TextMobject(r"$P_0(x_0,y_0,z_0)$").move_to(np.array([0,2.1,2.2])).set_color(YELLOW_C).scale(0.6)
+ #dot2_line = Line(np.array([0.8,0.8,0]), np.array([1,0.6,0]), color = PINK)
+
+ positive_vector = Arrow(np.array([1,1,0]), np.array([0.5,0.5,0]), buff=0.001, color = BLUE_C)
+ positive_gradient = Arrow(np.array([1,1,1.732]), np.array([0.5,0.5,1.9362]), buff=0.001, color = BLUE_C)
+ positive_gradient_lab = TextMobject(r"$\nabla f$").move_to(np.array([0.5,0.3,0])).set_color(BLUE_C).scale(0.5)
+
+ negative_vector = Arrow(np.array([1,1,0]), np.array([1.5,1.5,0]), buff=0.001, color = RED_C)
+ negative_gradient = Arrow(np.array([1,1,1.732]), np.array([1.5,1.5,1.322]), buff=0.001, color = RED_C)
+ negative_gradient_lab = TextMobject(r"$-\nabla f$").move_to(np.array([1.6,1.6,0])).set_color(RED_C).scale(0.5)
+
+ positive_vector_line = DashedLine(np.array([0.8,0.8,0]), np.array([1,-2,0]), color = WHITE)
+ positive_vector_lab = TextMobject(r"Most Rapid increase in $f$").move_to(np.array([1.6,-3.6,0])).set_color(BLUE_C).scale(0.6)
+ negative_vector_line = DashedLine(np.array([1.2,1.2,0]), np.array([3,-1.5,0]), color = WHITE)
+ negative_vector_lab = TextMobject(r"Most Rapid decrease in $f$").move_to(np.array([3.6,-3,0])).set_color(RED_C).scale(0.6)
+
+
+
+ line1 = DashedLine(np.array([0.5,0.5,0]), np.array([0.5,0.5,1.9362]), color = BLUE_C)
+ line2 = DashedLine(np.array([1,1,0]), np.array([1,1,1.732]), color = YELLOW_C)
+ line3 = DashedLine(np.array([1.5,1.5,0]), np.array([1.5,1.5,1.322]), color = RED_C)
+
+ curve_vector1 = Arrow(np.array([1,1,0]), np.array([1.5,0.5,0]), buff=0.001, color = YELLOW_C)
+ curve_vector2 = Arrow(np.array([1,1,0]), np.array([0.5,1.5,0]), buff=0.001, color = YELLOW_C)
+
+ curve_vector1_line = DashedLine(np.array([1.2,0.8,0]), np.array([1,2.5,0]), color = WHITE)
+ curve_vector2_line = DashedLine(np.array([0.8,1.2,0]), np.array([1,2.5,0]), color = WHITE)
+ curve_vector_lab = TextMobject(r"Zero Change in $f$").move_to(np.array([0.7,3.6,0])).set_color(PINK).scale(0.6)
+
+ #square = Square(side_length = 0.5).rotate(45*DEGREES).move_to(np.array([1.025,0.975,0]))
+ line_x = Line(np.array([0.8,0.8,0]), np.array([1,0.6,0]), color = PINK)
+ line_y = Line(np.array([1.2,0.8,0]), np.array([1,0.6,0]), color = PINK)
+
+ ninety_degree = VGroup(line_x, line_y)
+
+ self.set_camera_orientation(phi=60* DEGREES, theta = 20*DEGREES)
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.play(ShowCreation(quadrant))
+ self.wait()
+ self.play(ShowCreation(dot1), ShowCreation(dot2))
+ self.wait()
+ self.play(GrowArrow(positive_vector), GrowArrow(positive_gradient))
+ self.wait()
+ self.play(GrowArrow(negative_vector), GrowArrow(negative_gradient))
+ self.wait()
+ self.play(GrowArrow(line1), GrowArrow(line2), GrowArrow(line3))
+ self.wait()
+ self.play(ShowCreation(quadrant_curve))
+ self.wait()
+ self.play(GrowArrow(curve_vector1), GrowArrow(curve_vector2), ShowCreation(ninety_degree))
+ self.wait()
+ self.play(GrowArrow(dot1_line))
+ self.add_fixed_orientation_mobjects(dot1_lab)
+ self.wait()
+ self.play(GrowArrow(curve_vector1_line), GrowArrow(curve_vector2_line))
+ self.add_fixed_orientation_mobjects(curve_vector_lab)
+ self.wait()
+ self.add_fixed_orientation_mobjects(positive_gradient_lab, negative_gradient_lab)
+ self.wait()
+ self.play(GrowArrow(positive_vector_line), GrowArrow(negative_vector_line))
+ self.add_fixed_orientation_mobjects(positive_vector_lab, negative_vector_lab)
+ self.begin_ambient_camera_rotation(rate=0.1)
+ self.wait(3)
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/file3_gradient_level_curves.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/file3_gradient_level_curves.py new file mode 100644 index 0000000..a3b88e5 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/file3_gradient_level_curves.py @@ -0,0 +1,107 @@ +from manimlib.imports import *
+
+class GradientLevelCurves(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ paraboloid = ParametricSurface(
+ lambda u, v: np.array([
+ u*np.cos(v),
+ u*np.sin(v),
+ -u*u+2
+ ]),u_min=-1.414,u_max=1.414,v_min=0,v_max=2*PI, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1,
+ resolution=(15, 32)).scale(1)
+
+ plane_0 = Polygon(np.array([2,-2,0]),np.array([2,2,0]),np.array([-2,2,0]),np.array([-2,-2,0]),np.array([2,-2,0]), color = BLUE_E, fill_color = BLUE_E, fill_opacity = 0.3)
+ plane_0_lab = TextMobject("C = 0").move_to(0.4*UP+3.2*RIGHT).set_color(BLUE_E).scale(0.6)
+ circle_0 = Circle(radius = 1.414 , color = BLUE_E)
+ circle_0_lab = TextMobject("0").move_to(1.1*DOWN+1.1*RIGHT).set_color(BLUE_E).scale(0.6)
+
+ plane_0_5 = Polygon(np.array([2,-2,0.5]),np.array([2,2,0.5]),np.array([-2,2,0.5]),np.array([-2,-2,0.5]),np.array([2,-2,0.5]), color = GREEN_C, fill_color = GREEN_C, fill_opacity = 0.3)
+ plane_0_5_lab = TextMobject("C = 0.5").move_to(0.8*UP+3.4*RIGHT).set_color(GREEN_C).scale(0.6)
+ circle_0_5 = Circle(radius = 1.224 , color = GREEN_C)
+ circle_0_5_lab = TextMobject("0.5").move_to(0.9*DOWN+0.9*RIGHT).set_color(GREEN_C).scale(0.6)
+ circle_0_5_copy = circle_0_5.copy().move_to(np.array([0,0,0.5]))
+
+ plane_1 = Polygon(np.array([2,-2,1]),np.array([2,2,1]),np.array([-2,2,1]),np.array([-2,-2,1]),np.array([2,-2,1]), color = YELLOW_C, fill_color = YELLOW_C, fill_opacity = 0.3)
+ plane_1_lab = TextMobject("C = 1").move_to(1.2*UP+3.3*RIGHT).set_color(YELLOW_C).scale(0.6)
+ circle_1 = Circle(radius = 1 , color = YELLOW_C)
+ circle_1_lab = TextMobject("1").move_to(0.7*DOWN+0.7*RIGHT).set_color(YELLOW_C).scale(0.6)
+ circle_1_copy = circle_1.copy().move_to(np.array([0,0,1]))
+
+ plane_1_5 = Polygon(np.array([2,-2,1.5]),np.array([2,2,1.5]),np.array([-2,2,1.5]),np.array([-2,-2,1.5]),np.array([2,-2,1.5]), color = ORANGE, fill_color = ORANGE, fill_opacity = 0.3)
+ plane_1_5_lab = TextMobject("C = 1.5").move_to(1.7*UP+3.4*RIGHT).set_color(ORANGE).scale(0.6)
+ circle_1_5 = Circle(radius = 0.707 , color = ORANGE)
+ circle_1_5_lab = TextMobject("1.5").move_to(0.5*DOWN+0.5*RIGHT).set_color(ORANGE).scale(0.6)
+ circle_1_5_copy = circle_1_5.copy().move_to(np.array([0,0,1.5]))
+
+ plane_2 = Polygon(np.array([2,-2,2]),np.array([2,2,2]),np.array([-2,2,2]),np.array([-2,-2,2]),np.array([2,-2,2]), color = RED_C, fill_color = RED_C, fill_opacity = 0.3)
+ plane_2_lab = TextMobject("C = 2").move_to(2.1*UP+3.3*RIGHT).set_color(RED_C).scale(0.6)
+ dot_2 = Dot().set_fill(RED_C)
+ circle_2_lab = TextMobject("2").move_to(0.2*DOWN+0.2*RIGHT).set_color(RED_C).scale(0.6)
+ dot_2_copy = dot_2.copy().move_to(np.array([0,0,2]))
+
+ vector1 = Arrow(np.array([0.99,-0.99,0]), np.array([0.865,-0.865,0.5]), buff=0.01, color = RED_C).set_stroke(width=3)
+ gradient1 = Arrow(np.array([0.99,-0.99,0]), np.array([0.865,-0.865,0]), buff=0.01, color = RED_C).set_stroke(width=3)
+
+ vector2 = Arrow(np.array([0.865,-0.865,0.5]), np.array([0.707,-0.707,1]), buff=0.01, color = RED_C).set_stroke(width=3)
+ gradient2 = Arrow(np.array([0.865,-0.865,0]), np.array([0.707,-0.707,0]), buff=0.01, color = RED_C).set_stroke(width=3)
+
+ vector3 = Arrow(np.array([0.707,-0.707,1]), np.array([0.499,-0.499,1.5]), buff=0.01, color = RED_C).set_stroke(width=3)
+ gradient3 = Arrow(np.array([0.707,-0.707,0]), np.array([0.499,-0.499,0]), buff=0.01, color = RED_C).set_stroke(width=3)
+
+ vector4 = Arrow(np.array([0.499,-0.499,1.5]), np.array([0,0,2]), buff=0.01, color = RED_C).set_stroke(width=3)
+ gradient4 = Arrow(np.array([0.499,-0.499,0]), np.array([0,0,0]), buff=0.01, color = RED_C).set_stroke(width=3)
+
+
+ self.set_camera_orientation(phi=80 * DEGREES, theta = 0*DEGREES)
+ #self.set_camera_orientation(phi=45 * DEGREES, theta = -20*DEGREES)
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.play(Write(paraboloid))
+ self.wait()
+ self.play(ShowCreation(plane_0), ShowCreation(circle_0))
+ self.add_fixed_in_frame_mobjects(plane_0_lab)
+ self.wait()
+ self.play(ShowCreation(plane_0_5), ShowCreation(circle_0_5_copy), ShowCreation(circle_0_5))
+ self.add_fixed_in_frame_mobjects(plane_0_5_lab)
+ self.wait()
+ self.play(ShowCreation(plane_1), ShowCreation(circle_1_copy), ShowCreation(circle_1))
+ self.add_fixed_in_frame_mobjects(plane_1_lab)
+ self.wait()
+ self.play(ShowCreation(plane_1_5), ShowCreation(circle_1_5_copy), ShowCreation(circle_1_5))
+ self.add_fixed_in_frame_mobjects(plane_1_5_lab)
+ self.wait()
+ self.play(ShowCreation(plane_2), ShowCreation(dot_2_copy), ShowCreation(dot_2))
+ self.add_fixed_in_frame_mobjects(plane_2_lab)
+ self.wait()
+ self.move_camera(phi=60 * DEGREES, theta = 30*DEGREES,run_time=3)
+ self.play(FadeOut(plane_0), FadeOut(plane_0_lab), FadeOut(plane_0_5), FadeOut(plane_0_5_lab), FadeOut(plane_1), FadeOut(plane_1_lab), FadeOut(plane_1_5), FadeOut(plane_1_5_lab), FadeOut(plane_2), FadeOut(plane_2_lab))
+ self.play(FadeOut(circle_0_5_copy), FadeOut(circle_1_copy), FadeOut(circle_1_5_copy), FadeOut(dot_2_copy))
+
+ self.move_camera(phi=45 * DEGREES, theta = -20*DEGREES,run_time=3)
+ self.play(Write(vector1), Write(gradient1))
+ self.wait()
+ self.play(Write(vector2), Write(gradient2))
+ self.wait()
+ self.play(Write(vector3), Write(gradient3))
+ self.wait()
+ self.play(Write(vector4), Write(gradient4))
+ self.wait()
+ self.move_camera(phi=0 * DEGREES, theta = 0*DEGREES,run_time=3)
+ self.play(FadeOut(paraboloid))
+ self.play(FadeOut(vector1), FadeOut(vector2), FadeOut(vector3), FadeOut(vector4))
+ self.wait()
+ self.add_fixed_in_frame_mobjects(circle_0_lab, circle_0_5_lab, circle_1_lab, circle_1_5_lab,circle_2_lab)
+ self.wait(4)
+
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file1_directional_deriv.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file1_directional_deriv.gif Binary files differnew file mode 100644 index 0000000..39305d5 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file1_directional_deriv.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file2_gradient.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file2_gradient.gif Binary files differnew file mode 100644 index 0000000..d96f330 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file2_gradient.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file3_gradient_level_curves.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file3_gradient_level_curves.gif Binary files differnew file mode 100644 index 0000000..f1bf06a --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file3_gradient_level_curves.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/Multivariable_Functions_Quiz.pdf b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/Multivariable_Functions_Quiz.pdf Binary files differindex 7895843..7895843 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/Multivariable_Functions_Quiz.pdf +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/Multivariable_Functions_Quiz.pdf diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file1_multivar_func_examples.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file1_multivar_func_examples.py new file mode 100644 index 0000000..55b2b7e --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file1_multivar_func_examples.py @@ -0,0 +1,167 @@ +from manimlib.imports import *
+
+class Examples1(GraphScene):
+ def construct(self):
+
+ rectangle = Rectangle(height = 3, width = 4, color = GREEN)
+ rectangle_area_func = TexMobject("Area", "=", "f(", "Length", ",", "Breadth", ")").scale(0.6)
+ rectangle_area_func[0].set_color(RED_C)
+ rectangle_area_func[2].set_color(ORANGE)
+ rectangle_area_func[3].set_color(YELLOW_C)
+ rectangle_area_func[5].set_color(BLUE_C)
+ rectangle_area_func[6].set_color(ORANGE)
+
+
+ rectangle_area = TexMobject("Area", "=", "Length", "\\times", "Breadth").scale(0.6)
+ rectangle_area[0].set_color(RED_C)
+ rectangle_area[2].set_color(YELLOW_C)
+ rectangle_area[4].set_color(BLUE_C)
+
+
+ square = Square(side_length = 5, color = PURPLE)
+ square_area_func = TexMobject("Area", "=", "f(", "Length", ")")
+ square_area_func[0].set_color(GREEN_C)
+ square_area_func[2].set_color(ORANGE)
+ square_area_func[3].set_color(BLUE_C)
+ square_area_func[4].set_color(ORANGE)
+
+ square_area = TexMobject("Area", "=", "Length^2")
+ square_area[0].set_color(GREEN_C)
+ square_area[2].set_color(BLUE_C)
+
+
+ circle = Circle(radius = 2, color = PINK)
+ circle_area_func = TexMobject("Area", "=", "f(", "r", ")")
+ circle_area_func[0].set_color(YELLOW_C)
+ circle_area_func[2].set_color(ORANGE)
+ circle_area_func[3].set_color(GREEN_C)
+ circle_area_func[4].set_color(ORANGE)
+
+ circle_area = TexMobject("Area", "=", "\\pi", "r^2")
+ circle_area[0].set_color(YELLOW_C)
+ circle_area[2].set_color(BLUE_C)
+ circle_area[3].set_color(GREEN_C)
+
+ radius = Line(ORIGIN,2*RIGHT, color = RED_C)
+
+
+
+ braces_rect1 = Brace(rectangle, LEFT)
+ eq_text1 = braces_rect1.get_text("Length").set_color(YELLOW_C)
+ braces_rect2 = Brace(rectangle, UP)
+ eq_text2 = braces_rect2.get_text("Breadth").set_color(BLUE_C)
+
+ braces_square = Brace(square, LEFT)
+ braces_square_text = braces_square.get_text("Length").set_color(BLUE_C)
+
+ radius_text = TexMobject("r", color = GREEN_C).next_to(radius,UP)
+
+
+
+ self.play(ShowCreation(rectangle))
+ self.wait(1)
+ self.play(GrowFromCenter(braces_rect1),Write(eq_text1),GrowFromCenter(braces_rect2),Write(eq_text2))
+ self.wait(1)
+ self.play(Write(rectangle_area_func))
+ self.wait(1)
+ self.play(Transform(rectangle_area_func, rectangle_area))
+ self.wait(1)
+ self.play(FadeOut(braces_rect1),FadeOut(eq_text1),FadeOut(braces_rect2),FadeOut(eq_text2),FadeOut(rectangle_area_func))
+
+
+ self.play(Transform(rectangle, square))
+ self.wait(1)
+ self.play(GrowFromCenter(braces_square),Write(braces_square_text))
+ self.wait(1)
+ self.play(Write(square_area_func))
+ self.wait(1)
+ self.play(Transform(square_area_func, square_area))
+ self.wait(1)
+ self.play(FadeOut(braces_square),FadeOut(braces_square_text),FadeOut(square_area_func))
+
+
+ self.play(Transform(rectangle, circle))
+ self.wait(1)
+ self.play(ShowCreation(radius),Write(radius_text))
+ self.wait(1)
+ self.play(FadeOut(radius_text),FadeOut(radius))
+ self.wait(1)
+ self.play(Write(circle_area_func))
+ self.wait(1)
+ self.play(Transform(circle_area_func, circle_area))
+ self.wait(1)
+ self.play(FadeOut(circle_area_func))
+
+
+
+class Examples2(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ rectangle_x_y_0 = Polygon(np.array([-1,-2,0]),np.array([-1,2,0]),np.array([1,2,0]),np.array([1,-2,0]),np.array([-1,-2,0]), color = RED_E, fill_color = RED_C, fill_opacity = 0.1)
+ rectangle_x_y_3 = Polygon(np.array([-1,-2,3]),np.array([-1,2,3]),np.array([1,2,3]),np.array([1,-2,3]),np.array([-1,-2,3]), color = RED_E, fill_color = RED_C, fill_opacity = 0.1)
+
+ rectangle_y_z_1 = Polygon(np.array([1,-2,3]),np.array([1,2,3]),np.array([1,2,0]),np.array([1,-2,0]),np.array([1,-2,3]), color = RED_E, fill_color = RED_C, fill_opacity = 0.1)
+ rectangle_y_z_minus_1 = Polygon(np.array([-1,-2,3]),np.array([-1,2,3]),np.array([-1,2,0]),np.array([-1,-2,0]),np.array([-1,-2,3]), color = RED_E, fill_color = RED_C, fill_opacity = 0.1)
+
+ rectangle_x_z_2 = Polygon(np.array([1,2,3]),np.array([-1,2,3]),np.array([-1,2,0]),np.array([1,2,0]),np.array([1,2,3]), color = RED_E, fill_color = RED_C, fill_opacity = 0.1)
+ rectangle_x_z_minus_2 = Polygon(np.array([1,-2,3]),np.array([-1,-2,3]),np.array([-1,-2,0]),np.array([1,-2,0]),np.array([1,-2,3]), color = RED_E, fill_color = RED_C, fill_opacity = 0.1)
+
+ box = VGroup(rectangle_x_y_0, rectangle_x_y_3, rectangle_y_z_1, rectangle_y_z_minus_1, rectangle_x_z_2, rectangle_x_z_minus_2)
+
+ braces_rectangle_x_y_0 = Line(np.array([1,2,0]), np.array([1,-2,0]), color = BLUE_C)
+ braces_rectangle_x_y_0_text = TextMobject("Length").set_color(BLUE_C).move_to(np.array([2,-1,0]))
+
+ braces_rectangle_y_z_1 = Line(np.array([1,2,0]), np.array([1,2,3]), color = YELLOW_C)
+ braces_rectangle_y_z_1_text = TextMobject("Height").set_color(YELLOW_C).move_to(np.array([2,3.8,2]))
+
+ braces_rectangle_x_z_2 = Line(np.array([1,2,3]), np.array([-1,2,3]), color = PURPLE)
+ braces_rectangle_x_z_2_text = TextMobject("Breadth").set_color(PURPLE).move_to(np.array([0,3.8,3.3]))
+
+ box_area_func = TexMobject("Area =", "f(", "Length", ",", "Breadth", ",", "Height", ")").move_to(4*LEFT+3.5*UP).scale(0.6)
+ box_area_func[0].set_color(GREEN_C)
+ box_area_func[1].set_color(ORANGE)
+ box_area_func[2].set_color(BLUE_C)
+ box_area_func[4].set_color(PURPLE)
+ box_area_func[6].set_color(YELLOW_C)
+ box_area_func[7].set_color(ORANGE)
+
+ box_area_func_2 = TexMobject("Area =", "Length", "\\times", "Breadth", "\\times", "Height").move_to(4*LEFT+3.5*UP).scale(0.6)
+ box_area_func_2[0].set_color(GREEN_C)
+ box_area_func_2[1].set_color(BLUE_C)
+ box_area_func_2[3].set_color(PURPLE)
+ box_area_func_2[5].set_color(YELLOW_C)
+
+
+ self.set_camera_orientation(phi=70 * DEGREES, theta = 45*DEGREES)
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(3.7*UP)
+
+ self.add_fixed_in_frame_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.play(ShowCreation(box), ShowCreation(braces_rectangle_x_y_0))
+ self.add_fixed_orientation_mobjects(braces_rectangle_x_y_0_text)
+ self.play(ShowCreation(braces_rectangle_y_z_1))
+ self.add_fixed_orientation_mobjects(braces_rectangle_y_z_1_text)
+ self.play(ShowCreation(braces_rectangle_x_z_2))
+ self.add_fixed_orientation_mobjects(braces_rectangle_x_z_2_text)
+ self.wait(2)
+
+ self.move_camera(phi=60* DEGREES,theta=80*DEGREES)
+ self.add_fixed_in_frame_mobjects(box_area_func)
+ self.play(Write(box_area_func))
+ self.wait()
+
+
+ self.play(ReplacementTransform(box_area_func,box_area_func_2))
+ self.add_fixed_in_frame_mobjects(box_area_func_2)
+
+
+ self.wait(3)
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file2_multivariable_func_respresentation.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file2_multivariable_func_respresentation.py new file mode 100644 index 0000000..d10ff0a --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file2_multivariable_func_respresentation.py @@ -0,0 +1,98 @@ +from manimlib.imports import *
+
+class MultivariableFunc(Scene):
+ def construct(self):
+
+ topic = TextMobject("Multivariable Functions")
+ topic.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+ topic.scale(1.5)
+
+ self.play(Write(topic))
+ self.wait()
+ self.play(FadeOut(topic))
+
+
+ #circle = Circle()
+ #circle.scale(3)
+
+ scalar_function = TextMobject("Scalar Valued Function")
+ scalar_function.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+ scalar_function.scale(1.5)
+ scalar_function.move_to(2.5*UP)
+
+ rectangle = Rectangle(height = 2, width = 4)
+ rectangle.set_color(PURPLE)
+
+ eqn1 = TextMobject(r"f(x,y) = $x^2y$")
+ eqn1.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE)
+
+
+
+ number1 = TextMobject("(2,1)")
+ number1.move_to(2.5*UP+ 4*LEFT)
+ number1.scale(1.2)
+ number1.set_color(ORANGE)
+
+ output1 = TextMobject("4")
+ output1.scale(1.5)
+ output1.set_color(BLUE_C)
+ output1.move_to(3*RIGHT)
+
+ eqn1_1 = TextMobject(r"f(2,1) = $2^2(1)$")
+ eqn1_1.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE)
+
+
+ self.play(Write(eqn1),ShowCreation(rectangle))
+ self.wait()
+ self.play(ApplyMethod(number1.move_to, 3*LEFT))
+ self.play(FadeOut(number1))
+ self.play(Transform(eqn1, eqn1_1))
+ self.wait()
+ self.play(ApplyMethod(output1.move_to, 2.5*DOWN+4*RIGHT))
+ self.wait()
+ self.play(Write(scalar_function))
+ self.play(FadeOut(output1), FadeOut(scalar_function), FadeOut(eqn1))
+
+
+ vector_function = TextMobject("Vector Valued Function")
+ vector_function.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+ vector_function.scale(1.5)
+ vector_function.move_to(2.5*UP)
+
+
+ eqn2 = TextMobject(r"f(x,y,z) = $ \begin{bmatrix} x^2y \\ 2yz \end{bmatrix}$")
+ eqn2.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+
+ number2 = TextMobject("(2,1,3)")
+ number2.move_to(2.5*UP+ 4*LEFT)
+ number2.scale(1.2)
+ number2.set_color(ORANGE)
+
+ output2 = TextMobject(r"$ \begin{bmatrix} 4 \\ 6 \end{bmatrix}$")
+ #output2.scale(1.5)
+ output2.set_color(BLUE_C)
+ output2.move_to(3*RIGHT)
+
+ #eqn2_1 = TextMobject(r"f(2,1,3) = $2^2(1) + 2(1)(3)$")
+ #eqn2_1.set_color(YELLOW)
+
+ #eqn2_2 = TextMobject(r"f(2,1,3) = $2 + 6$")
+ #eqn2_2.set_color(YELLOW)
+
+
+ self.play(Write(eqn2))
+
+ self.wait()
+ self.play(ApplyMethod(number2.move_to, 3*LEFT))
+ self.play(FadeOut(number2))
+
+ #self.play(Transform(eqn2, eqn2_1))
+ #self.wait(1)
+ #self.play(Transform(eqn2, eqn2_2))
+ #self.wait(1)
+
+ self.play(ApplyMethod(output2.move_to, 2.5*DOWN+4*RIGHT))
+ self.wait()
+ self.play(Write(vector_function))
+ self.play(FadeOut(output2),FadeOut(eqn2), FadeOut(vector_function), FadeOut(rectangle))
+ self.wait()
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file3_sphere.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file3_sphere.py new file mode 100644 index 0000000..86239ae --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file3_sphere.py @@ -0,0 +1,177 @@ +from manimlib.imports import *
+
+class Sphere(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes() # creates a 3D Axis
+
+ text3d = TextMobject(r"$f(x,y) \rightarrow Point(x,y,z)$")
+ text3d1 = TextMobject(r"$f(x,y) \rightarrow Point(x,y, \sqrt{r^2 - x^2 - y^2})$")
+ self.add_fixed_in_frame_mobjects(text3d)
+ text3d.scale(0.7)
+ text3d1.scale(0.7)
+ text3d.to_corner(UL)
+ text3d1.to_corner(UL)
+ text3d.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+ text3d1.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+ self.play(Write(text3d))
+ self.wait(1)
+
+ self.play(Transform(text3d,text3d1))
+ self.add_fixed_in_frame_mobjects(text3d1)
+ self.play(FadeOut(text3d))
+
+ sphere = ParametricSurface(
+ lambda u, v: np.array([
+ 2*np.sin(u)*np.cos(v),
+ 2*np.sin(u)*np.sin(v),
+ 2*np.cos(u)
+ ]),u_min=0,u_max=PI,v_min=0,v_max=2*PI,checkerboard_colors=[RED_D, RED_E],
+ resolution=(15, 32)).scale(1)
+
+
+ #Experiment with circles by changing difference value of u and v
+ '''
+ sphere_points = [np.array([2*np.sin(u*DEGREES)*np.cos(v*DEGREES), 2*np.sin(u*DEGREES)*np.sin(v*DEGREES), 2*np.cos(u*DEGREES)]) for u in range(0, 185, 5) for v in range(0, 365, 5)]
+
+ sphere_spheres = [Dot().move_to(pts) for pts in sphere_points]
+
+ sphere = VGroup(*sphere_spheres)
+ '''
+
+ self.set_camera_orientation(phi=75 * DEGREES, theta = 45*DEGREES)
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ dot_x_y1 = Dot().scale(0.75).set_fill(RED_C).move_to(np.array([-1,1,0]))
+ dot_x_y_z1 = Dot().scale(0.75).set_fill(RED_C).move_to(np.array([-1,1,1.414]))
+ dot_x_y_z_1 = Dot().scale(0.75).set_fill(RED_C).move_to(np.array([-1,1,-1.414]))
+ line1 = DashedLine(np.array([-1,1,-1.414]), np.array([-1,1,1.414]), color = YELLOW_C)
+
+ point_x_y1 = TexMobject("(-1,1,0)").set_color(BLUE_C).move_to(np.array([-1.5,1.5,0])).scale(0.5)
+ point_x_y_z1 = TexMobject("(-1,1,\\sqrt{r^2 - x^2 - y^2})").set_color(BLUE_C).move_to(np.array([-1.5,1.5,1.414])).scale(0.5)
+ point_x_y_z1_2 = TexMobject("(-1,1,\\sqrt{4 - x^2 - y^2})").set_color(BLUE_C).move_to(np.array([-1.5,1.5,1.414])).scale(0.5)
+ point_x_y_z1_3 = TexMobject("(-1,1,\\sqrt{4 - 1 - 1})").set_color(BLUE_C).move_to(np.array([-1.5,1.5,1.414])).scale(0.5)
+ point_x_y_z1_4 = TexMobject("(-1,1,\\sqrt{2})").set_color(BLUE_C).move_to(np.array([-1.5,1.5,1.414])).scale(0.5)
+ point_x_y_z1_5 = TexMobject("(-1,1,1.414)").set_color(BLUE_C).move_to(np.array([-1.5,1.5,1.414])).scale(0.5)
+
+ point_x_y_z_1 = TexMobject("(-1,1,\\sqrt{r^2 - x^2 - y^2})").set_color(BLUE_C).move_to(np.array([-1.5,1.5,-1.414])).scale(0.5)
+ point_x_y_z_1_2 = TexMobject("(-1,1,\\sqrt{4 - x^2 - y^2})").set_color(BLUE_C).move_to(np.array([-1.5,1.5,-1.414])).scale(0.5)
+ point_x_y_z_1_3 = TexMobject("(-1,1,\\sqrt{4 - 1 - 1})").set_color(BLUE_C).move_to(np.array([-1.5,1.5,-1.414])).scale(0.5)
+ point_x_y_z_1_4 = TexMobject("(-1,1,\\sqrt{2})").set_color(BLUE_C).move_to(np.array([-1.5,1.5,-1.414])).scale(0.5)
+ point_x_y_z_1_5 = TexMobject("(-1,1,-1.414)").set_color(BLUE_C).move_to(np.array([-1.5,1.5,-1.414])).scale(0.5)
+
+
+ self.play(ShowCreation(dot_x_y1))
+ self.add_fixed_orientation_mobjects(point_x_y1)
+ self.play(ShowCreation(dot_x_y_z1), ShowCreation(dot_x_y_z_1), ShowCreation(line1))
+ self.add_fixed_orientation_mobjects(point_x_y_z1, point_x_y_z_1)
+ self.wait(0.5)
+ self.play(ReplacementTransform(point_x_y_z1,point_x_y_z1_2), ReplacementTransform(point_x_y_z_1,point_x_y_z_1_2))
+ self.add_fixed_orientation_mobjects(point_x_y_z1_2, point_x_y_z_1_2)
+
+ self.wait(0.5)
+ self.play(ReplacementTransform(point_x_y_z1_2,point_x_y_z1_3), ReplacementTransform(point_x_y_z_1_2,point_x_y_z_1_3))
+ self.add_fixed_orientation_mobjects(point_x_y_z1_3, point_x_y_z_1_3)
+ self.wait(0.5)
+ self.play(ReplacementTransform(point_x_y_z1_3,point_x_y_z1_4), ReplacementTransform(point_x_y_z_1_3,point_x_y_z_1_4))
+ self.add_fixed_orientation_mobjects(point_x_y_z1_4, point_x_y_z_1_4)
+ self.wait(0.5)
+ self.play(ReplacementTransform(point_x_y_z1_4,point_x_y_z1_5), ReplacementTransform(point_x_y_z_1_4,point_x_y_z_1_5))
+ self.add_fixed_orientation_mobjects(point_x_y_z1_5, point_x_y_z_1_5)
+
+
+
+ dot_x_y2 = Dot().scale(0.75).set_fill(RED_C).move_to(np.array([0.5,-0.5,0]))
+ dot_x_y_z2 = Dot().scale(0.75).set_fill(RED_C).move_to(np.array([0.5,-0.5,1.87]))
+ dot_x_y_z_2 = Dot().scale(0.75).set_fill(RED_C).move_to(np.array([0.5,-0.5,-1.87]))
+ line2 = DashedLine(np.array([0.5,-0.5,-1.87]), np.array([0.5,-0.5,1.87]), color = YELLOW_C)
+
+ point_x_y2 = TexMobject("(0.5,-0.5,0)").set_color(BLUE_C).move_to(np.array([1.5,-1.5,0])).scale(0.5)
+ point_x_y_z2 = TexMobject("(0.5,-0.5,\\sqrt{r^2 - x^2 - y^2})").set_color(BLUE_C).move_to(np.array([1.5,-1.5,1.87])).scale(0.5)
+ point_x_y_z2_2 = TexMobject("(0.5,-0.5,\\sqrt{4 - x^2 - y^2})").set_color(BLUE_C).move_to(np.array([1.5,-1.5,1.87])).scale(0.5)
+ point_x_y_z2_3 = TexMobject("(0.5,-0.5,\\sqrt{4 - 0.25 - 0.25})").set_color(BLUE_C).move_to(np.array([1.5,-1.5,1.87])).scale(0.5)
+ point_x_y_z2_4 = TexMobject("(0.5,-0.5,\\sqrt{3.5})").set_color(BLUE_C).move_to(np.array([1.5,-1.5,1.87])).scale(0.5)
+ point_x_y_z2_5 = TexMobject("(0.5,-0.5,1.87)").set_color(BLUE_C).move_to(np.array([1.5,-1.5,1.87])).scale(0.5)
+
+ point_x_y_z_2 = TexMobject("(0.5,-0.5,\\sqrt{r^2 - x^2 - y^2})").set_color(BLUE_C).move_to(np.array([1.5,-1.5,-1.87])).scale(0.5)
+ point_x_y_z_2_2 = TexMobject("(0.5,-0.5,\\sqrt{4 - x^2 - y^2})").set_color(BLUE_C).move_to(np.array([1.5,-1.5,-1.87])).scale(0.5)
+ point_x_y_z_2_3 = TexMobject("(0.5,-0.5,\\sqrt{4 - 0.25 - 0.25})").set_color(BLUE_C).move_to(np.array([1.5,-1.5,-1.87])).scale(0.5)
+ point_x_y_z_2_4 = TexMobject("(0.5,-0.5,\\sqrt{3.5})").set_color(BLUE_C).move_to(np.array([1.5,-1.5,-1.87])).scale(0.5)
+ point_x_y_z_2_5 = TexMobject("(0.5,-0.5,-1.87)").set_color(BLUE_C).move_to(np.array([1.5,-1.5,-1.87])).scale(0.5)
+
+
+ self.play(ShowCreation(dot_x_y2))
+ self.add_fixed_orientation_mobjects(point_x_y2)
+ self.play(ShowCreation(dot_x_y_z2), ShowCreation(dot_x_y_z_2), ShowCreation(line2))
+ self.add_fixed_orientation_mobjects(point_x_y_z2, point_x_y_z_2)
+ self.wait(0.5)
+ self.play(ReplacementTransform(point_x_y_z2,point_x_y_z2_2), ReplacementTransform(point_x_y_z_2,point_x_y_z_2_2))
+ self.add_fixed_orientation_mobjects(point_x_y_z2_2, point_x_y_z_2_2)
+
+ self.wait(0.5)
+ self.play(ReplacementTransform(point_x_y_z2_2,point_x_y_z2_3), ReplacementTransform(point_x_y_z_2_2,point_x_y_z_2_3))
+ self.add_fixed_orientation_mobjects(point_x_y_z2_3, point_x_y_z_2_3)
+ self.wait(0.5)
+ self.play(ReplacementTransform(point_x_y_z2_3,point_x_y_z2_4), ReplacementTransform(point_x_y_z_2_3,point_x_y_z_2_4))
+ self.add_fixed_orientation_mobjects(point_x_y_z2_4, point_x_y_z_2_4)
+ self.wait(0.5)
+ self.play(ReplacementTransform(point_x_y_z2_4,point_x_y_z2_5), ReplacementTransform(point_x_y_z_2_4,point_x_y_z_2_5))
+ self.add_fixed_orientation_mobjects(point_x_y_z2_5, point_x_y_z_2_5)
+
+ self.play(FadeOut(point_x_y1), FadeOut(point_x_y_z1_5), FadeOut(point_x_y_z_1_5), FadeOut(dot_x_y1), FadeOut(dot_x_y_z1), FadeOut(dot_x_y_z_1), FadeOut(line1))
+ self.play(FadeOut(point_x_y2), FadeOut(point_x_y_z2_5), FadeOut(point_x_y_z_2_5), FadeOut(dot_x_y2), FadeOut(dot_x_y_z2), FadeOut(dot_x_y_z_2), FadeOut(line2))
+
+
+
+
+ sphere_final = []
+
+ for u in range(0, 180, 15):
+ sphere_points1 = [np.array([2*np.sin(u*DEGREES)*np.cos(v*DEGREES), 2*np.sin(u*DEGREES)*np.sin(v*DEGREES), 2*np.cos(u*DEGREES)]) for v in range(0, 370, 10)]
+ sphere_dots1 = [Dot().scale(0.75).set_fill(RED_C).move_to(pts) for pts in sphere_points1]
+
+ sphere_points2 = [np.array([2*np.sin((u+5)*DEGREES)*np.cos(v*DEGREES), 2*np.sin((u+5)*DEGREES)*np.sin(v*DEGREES), 2*np.cos((u+5)*DEGREES)]) for v in range(0, 370, 10)]
+ sphere_dots2 = [Dot().scale(0.75).set_fill(RED_C).move_to(pts) for pts in sphere_points2]
+
+ sphere_points3 = [np.array([2*np.sin((u+10)*DEGREES)*np.cos(v*DEGREES), 2*np.sin((u+10)*DEGREES)*np.sin(v*DEGREES), 2*np.cos((u+10)*DEGREES)]) for v in range(0, 370, 10)]
+ sphere_dots3 = [Dot().scale(0.75).set_fill(RED_C).move_to(pts) for pts in sphere_points3]
+
+ sphere_final = sphere_final + sphere_dots1 + sphere_dots2 + sphere_dots3
+
+ sphere_dots = sphere_dots1 + sphere_dots2 + sphere_dots3
+
+ sphere_with_dots = VGroup(*sphere_dots)
+ self.play(ShowCreation(sphere_with_dots))
+
+ sphere_final_with_dots = VGroup(*sphere_final)
+
+
+ self.begin_ambient_camera_rotation(rate=0.5)
+ self.wait(3)
+ self.play(ReplacementTransform(sphere_final_with_dots, sphere))
+ self.wait(5)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_vectorvf_sine.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file4_vectorvf_sine.py index 06e225e..06e225e 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_vectorvf_sine.py +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file4_vectorvf_sine.py diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file5_vectorvf_helix.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file5_vectorvf_helix.py new file mode 100644 index 0000000..fc151ac --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file5_vectorvf_helix.py @@ -0,0 +1,92 @@ +from manimlib.imports import *
+
+class Helix(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes() # creates a 3D Axis
+
+ helix1=ParametricFunction(
+ lambda u : np.array([
+ 1.5*np.cos(u),
+ 1.5*np.sin(u),
+ u/4
+ ]),color=PURPLE,t_min=-TAU,t_max=TAU,
+ )
+
+ helix2=ParametricFunction(
+ lambda u : np.array([
+ 2*np.cos(u),
+ 2*np.sin(u),
+ u/2
+ ]),color=GREEN_C,t_min=-TAU,t_max=TAU,
+ )
+
+ function = TexMobject("f(", "r", ",", "\\theta", ")", "=", "[", "r", "\\cos", "\\theta", ",", "r", "\\sin" ,"\\theta", ",", "h" ,"\\theta", "]" ).scale(0.6).to_corner(UL)
+ function.set_color_by_tex(r"\theta", BLUE_C)
+ function.set_color_by_tex(r"r", RED_C)
+ function.set_color_by_tex(r"\cos", GREEN_C)
+ function.set_color_by_tex(r"\sin", YELLOW_C)
+ function[0].set_color(ORANGE)
+ function[4].set_color(ORANGE)
+
+
+ self.add_fixed_in_frame_mobjects(function)
+
+ self.set_camera_orientation(phi=60*DEGREES, theta = 45*DEGREES)
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+
+ dot1 = Dot().rotate(PI/2).set_color(RED_C)
+ alpha1 = ValueTracker(0)
+ vector1 = self.get_vector(alpha1.get_value(),helix1)
+ dot1.add_updater(lambda m: m.move_to(vector1.get_end()))
+ self.play(
+ ShowCreation(helix1),
+ GrowFromCenter(dot1),
+ GrowArrow(vector1)
+ )
+ vector1.add_updater(
+ lambda m: m.become(
+ self.get_vector(alpha1.get_value()%1,helix1)
+ )
+ )
+ self.add(vector1,dot1)
+ self.play(alpha1.increment_value, 1, run_time=10, rate_func=linear)
+
+
+ self.play(FadeOut(vector1), FadeOut(dot1))
+ self.play(ReplacementTransform(helix1, helix2))
+
+
+ dot2 = Dot().rotate(PI/2).set_color(RED_C)
+ alpha2 = ValueTracker(0)
+ vector2 = self.get_vector(alpha2.get_value(),helix2)
+ dot2.add_updater(lambda m: m.move_to(vector2.get_end()))
+ self.play(
+ ShowCreation(helix2),
+ GrowFromCenter(dot2),
+ GrowArrow(vector2)
+ )
+ vector2.add_updater(
+ lambda m: m.become(
+ self.get_vector(alpha2.get_value()%1,helix2)
+ )
+ )
+ self.add(vector2,dot2)
+ self.play(alpha2.increment_value, 1, run_time=10, rate_func=linear)
+ self.wait()
+
+
+
+ def get_vector(self, proportion, curve):
+ vector = Line(np.array([0,0,0]), curve.point_from_proportion(proportion), color = YELLOW_C, buff=0)
+ return vector
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_derivative_vectorvf.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file6_derivative_vectorvf.py index 466e389..466e389 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/multivariable_func_derivative_vectorvf.py +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/file6_derivative_vectorvf.py diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file1_multivar_func_examples.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file1_multivar_func_examples.gif Binary files differnew file mode 100644 index 0000000..43c3a42 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file1_multivar_func_examples.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file2_multivariable_func_respresentation.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file2_multivariable_func_respresentation.gif Binary files differnew file mode 100644 index 0000000..40add0f --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file2_multivariable_func_respresentation.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file3_sphere.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file3_sphere.gif Binary files differnew file mode 100644 index 0000000..3e35ec8 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file3_sphere.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_vectorvf_sine.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file4_vectorvf_sine.gif Binary files differindex 4f6b931..4f6b931 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_vectorvf_sine.gif +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file4_vectorvf_sine.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file5_vectorvf_helix.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file5_vectorvf_helix.gif Binary files differnew file mode 100644 index 0000000..c3d37f6 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file5_vectorvf_helix.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_derivative_vectorvf.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file6_derivative_vectorvf.gif Binary files differindex a94de90..a94de90 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Multivariable Functions/gifs/multivariable_func_derivative_vectorvf.gif +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file6_derivative_vectorvf.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/Limits_and_Continuity_of_Multivariable_Function_Quiz.pdf b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/Limits_and_Continuity_of_Multivariable_Function_Quiz.pdf Binary files differnew file mode 100644 index 0000000..99918e5 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/Limits_and_Continuity_of_Multivariable_Function_Quiz.pdf diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file1_epsilon_delta_defn.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file1_epsilon_delta_defn.py new file mode 100644 index 0000000..803c122 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file1_epsilon_delta_defn.py @@ -0,0 +1,179 @@ +from manimlib.imports import *
+
+class EpsilonDelta(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes() # creates a 3D Axis
+
+
+ sphere = ParametricSurface(
+ lambda u, v: np.array([
+ 3*np.sin(u)*np.cos(v),
+ 3*np.sin(u)*np.sin(v),
+ 3*np.cos(u)
+ ]),u_min=0,u_max=PI/4,v_min=PI/2,v_max=PI,checkerboard_colors=[RED_D, RED_E],
+ resolution=(15, 32)).scale(1)
+
+
+ cylinder_z = ParametricSurface(
+ lambda u, v: np.array([
+ 0.25*np.cos(TAU * v),
+ 1.8* (1 - u),
+ 0.25*np.sin(TAU * v)
+
+ ]),
+ checkerboard_colors=[YELLOW_C, YELLOW_E], resolution=(6, 32)).fade(0.2).rotate(PI/4).move_to(np.array([-0.65,0.65,2.54]))
+
+
+ cylinder_x = ParametricSurface(
+ lambda u, v: np.array([
+ 0.3*np.cos(TAU * v)-1,
+ 0.3*np.sin(TAU * v)+1,
+ 2.6*(1 - u)
+ ]),
+ checkerboard_colors=[BLUE_C, BLUE_E], resolution=(6, 32)).fade(0.2)
+
+
+ delta_circle = Circle(radius= 0.3, color = BLACK).shift(1*LEFT+1*UP).set_fill(GREEN_E, opacity = 0.5)
+
+ epsilon_circle = [np.array([0.25*np.cos(i*DEGREES),0,0.25*np.sin(i*DEGREES)]) for i in range(361)]
+
+ epsilon_circle_polygon = Polygon(*epsilon_circle, color = RED_E, fill_color = RED_E, fill_opacity = 0.5).rotate(PI/4).move_to(np.array([0,0,2.54]))
+
+
+ dot_circle = Dot().move_to(np.array([-1,1,0])).set_fill("#000080")
+
+ dot_surface = Dot().rotate(-PI/4).scale(1.5).move_to(np.array([-1.2,1.2,2.7])).set_fill("#000080")
+
+ dot_L_epsilon1 = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).rotate(PI/4).move_to(np.array([0,0,2.3]))
+
+ dot_L_epsilon2 = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).rotate(PI/4).move_to(np.array([0,0,2.8]))
+
+ dot_L = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#006400", fill_color = "#006400", fill_opacity = 1).rotate(PI/4).move_to(np.array([0,0,2.54]))
+
+
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.set_camera_orientation(phi=75*DEGREES,theta=135*DEGREES)
+ #self.set_camera_orientation(phi=80*DEGREES,theta=45*DEGREES)
+
+
+ self.play(ShowCreation(sphere),ShowCreation(delta_circle), ShowCreation(dot_circle))
+
+ temp_circle_center = TextMobject(r"$(a,b,0)$").scale(0.6).set_color(BLUE_C).move_to(1.7*LEFT+1.1*UP)
+ self.add_fixed_orientation_mobjects(temp_circle_center)
+ self.wait()
+
+ delta_lab = TextMobject(r"$\delta$", r"$-$", "disk").scale(0.5).move_to(0.6*LEFT+1.7*UP)
+ delta_lab[0].set_color(PINK).scale(1.3)
+ delta_lab[1].set_color(ORANGE)
+ delta_lab[2].set_color(GREEN_E)
+
+ self.add_fixed_orientation_mobjects(delta_lab)
+
+ self.play(ShowCreation(dot_surface))
+
+ temp_curve_circle_center = TextMobject(r"$(a,b,L)$").scale(0.6).set_color("#006400").move_to(np.array([-2,1,2.7]))
+ self.add_fixed_orientation_mobjects(temp_curve_circle_center)
+
+
+ self.wait()
+ self.play(ShowCreation(cylinder_x), FadeOut(dot_surface))
+ self.wait()
+
+ self.move_camera(phi=0* DEGREES,theta=135*DEGREES)
+ self.wait()
+
+ self.move_camera(phi=80* DEGREES,theta=225*DEGREES)
+ self.wait()
+
+ self.play(FadeOut(delta_lab), ShowCreation(cylinder_z))
+ self.wait()
+
+ self.play(FadeOut(temp_circle_center), FadeOut(temp_curve_circle_center),ShowCreation(epsilon_circle_polygon))
+
+ self.move_camera(phi=80* DEGREES,theta=325*DEGREES)
+
+ dot_L_epsilon1_lab = TextMobject(r"$L$", r"$-$", r"$\epsilon$").scale(0.6).move_to(np.array([-0.4,-0.4,2.3]))
+ dot_L_epsilon1_lab[0].set_color("#D4108A")
+ dot_L_epsilon1_lab[1].set_color("#006400")
+ dot_L_epsilon1_lab[2].set_color("#4DC8A1").scale(1.5)
+
+ dot_L_epsilon2_lab = TextMobject(r"$L$", r"$+$", r"$\epsilon$").scale(0.6).move_to(np.array([-0.4,-0.4,2.8]))
+ dot_L_epsilon2_lab[0].set_color("#D4108A")
+ dot_L_epsilon2_lab[1].set_color("#006400")
+ dot_L_epsilon2_lab[2].set_color("#4DC8A1").scale(1.5)
+
+ dot_L_lab = TextMobject(r"$L$").scale(0.6).set_color("#D4108A").move_to(np.array([-0.4,-0.4,2.54]))
+
+
+ self.play(ShowCreation(dot_L_epsilon1), ShowCreation(dot_L), ShowCreation(dot_L_epsilon2))
+ self.add_fixed_orientation_mobjects(dot_L_epsilon1_lab, dot_L_epsilon2_lab, dot_L_lab)
+ self.wait(4)
+
+ self.move_camera(phi=80* DEGREES,theta=45*DEGREES)
+ self.wait(2)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ '''
+
+
+
+
+
+
+
+
+
+
+
+ delta_lab = TextMobject(r"$\delta - disk$")
+ delta_lab.scale(0.5)
+ delta_lab.set_color(PINK)
+
+ self.play(ShowCreation(circle_center))
+ self.add_fixed_in_frame_mobjects(temp_circle_center)
+ temp_circle_center.move_to(1.5*RIGHT)
+ self.play(Write(temp_circle_center))
+
+ self.play(ShowCreation(curve_circle_center))
+ self.add_fixed_in_frame_mobjects(temp_curve_circle_center)
+ temp_curve_circle_center.move_to(1.9*UP+1*RIGHT)
+ self.play(Write(temp_curve_circle_center))
+
+
+ self.add_fixed_in_frame_mobjects(delta_lab)
+ delta_lab.move_to(0.4*DOWN+1.7*RIGHT)
+ self.play(Write(delta_lab))
+
+
+
+
+
+ self.begin_ambient_camera_rotation(rate=0.2)
+
+ self.play(ShowCreation(circle), ShowCreation(line1), ShowCreation(line2))
+ self.play(ShowCreation(line3), ShowCreation(line4))
+ self.wait(8)
+ '''
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Limits and Continuity of Multivariable Functions/limit_approach_point.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file2_limit_approach_point.py index 57d1d45..57d1d45 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Limits and Continuity of Multivariable Functions/limit_approach_point.py +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file2_limit_approach_point.py diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file3_limit_approach_point_3d.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file3_limit_approach_point_3d.py new file mode 100644 index 0000000..f1007a4 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file3_limit_approach_point_3d.py @@ -0,0 +1,152 @@ +from manimlib.imports import *
+
+class Limit(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ text3d = TextMobject(r"$f(x,y) = \frac{x - y}{x - 1}$")
+ self.add_fixed_in_frame_mobjects(text3d)
+
+ text3d.to_corner(UL)
+
+ text3d.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+
+ self.play(Write(text3d))
+ self.wait(1)
+
+ limit_func = ParametricSurface(
+ lambda u, v: np.array([
+ 3*np.sin(u)*np.cos(v),
+ 3*np.sin(u)*np.sin(v),
+ (3*np.sin(u)*np.cos(v) - 3*np.sin(u)*np.sin(v))/2*(3*np.sin(u)*np.cos(v) - 1)
+ ]),u_min=0,u_max=PI,v_min=0,v_max=2*PI, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1,
+ resolution=(15, 32)).scale(1)
+
+ limit_y_x =ParametricFunction(
+ lambda u : np.array([
+ u,
+ u,
+ 0
+ ]),color=GREEN_D,t_min=-3,t_max=3,
+ )
+
+ limit_y_1 =ParametricFunction(
+ lambda u : np.array([
+ u,
+ 1,
+ 1/2
+ ]),color=BLUE_D,t_min=-3,t_max=3,
+ )
+
+ limit_y_x_2 =ParametricFunction(
+ lambda u : np.array([
+ u,
+ u*u,
+ (u - u*u)/2*(u - 1)
+ ]),color=RED_D,t_min=-3,t_max=3,
+ )
+
+ limit_y_2_x =ParametricFunction(
+ lambda u : np.array([
+ u,
+ 2 - u,
+ 1
+ ]),color=YELLOW_D,t_min=-3,t_max=3,
+ )
+
+ plane_y_x = Polygon(np.array([-3,-3,-3]),np.array([3,3,-3]),np.array([3,3,3]),np.array([-3,-3,3]),np.array([-3,-3,-3]), color = GREEN_C, fill_color = GREEN_C, fill_opacity = 0.1)
+ plane_y_x_text = TextMobject(r"$y = x$", color = GREEN_C).move_to(np.array([5,0,3]))
+
+ plane_y_1 = Polygon(np.array([-3,1,-3]),np.array([3,1,-3]),np.array([3,1,3]),np.array([-3,1,3]),np.array([-3,1,-3]), color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1)
+ plane_y_1_text = TextMobject(r"$y = 1$", color = BLUE_C).move_to(np.array([5,0,2.5]))
+
+
+ #Creating plane y = x^2
+ ######
+ y_x_2 = []
+ y_x_2.append(np.array([2, 4, -3]))
+ y_x_2.append(np.array([2, 4, 3]))
+ y_x_2_1 = [np.array([i, i*i, 3]) for i in np.arange(1.9,-2.1, -0.1)]
+
+ y_x_2 = y_x_2 + y_x_2_1
+
+ y_x_2.append(np.array([-2, 4, 3]))
+ y_x_2.append(np.array([-2, 4, -3]))
+
+ y_x_2_2 = [np.array([i, i*i, -3]) for i in np.arange(-2,2.1, 0.1)]
+
+ y_x_2 = y_x_2 + y_x_2_2
+ #y_x_2.append(np.array([-3, 9, 0]))
+
+ plane_y_x_2 = Polygon(*y_x_2, color = RED_C, fill_color = RED_C, fill_opacity = 0.1)
+ plane_y_x_2_text = TextMobject(r"$y = x^2$", color = RED_C).move_to(np.array([5,0,2]))
+
+ ######
+
+ plane_y_2_x = Polygon(np.array([-3,5,-3]),np.array([3,-1,-3]),np.array([3,-1,3]),np.array([-3,5,3]),np.array([-3,5,-3]), color = YELLOW_C, fill_color = YELLOW_C, fill_opacity = 0.1)
+ plane_y_2_x_text = TextMobject(r"$y = 2 - x$", color = YELLOW_C).move_to(np.array([5,0,1.5]))
+
+ line_1_1 = Line(np.array([1,1,-3]), np.array([1,1,3]), color = PINK)
+
+ point = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).move_to(np.array([1,1,0]))
+ point_text = TextMobject(r"$(1,1,0)$", color = WHITE).scale(0.7).move_to(np.array([1.8,1,0]))
+
+
+
+
+ self.set_camera_orientation(phi=70 * DEGREES, theta = -95*DEGREES)
+
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(3.7*UP)
+
+ self.add_fixed_in_frame_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.play(ShowCreation(limit_func))
+ self.wait(2)
+
+ self.play(ShowCreation(plane_y_x))
+ self.add_fixed_orientation_mobjects(plane_y_x_text)
+ self.play(ShowCreation(limit_y_x))
+ self.wait()
+
+ self.play(ShowCreation(plane_y_1))
+ self.add_fixed_orientation_mobjects(plane_y_1_text)
+ self.play(ShowCreation(limit_y_1))
+ self.wait()
+
+ self.play(ShowCreation(plane_y_x_2))
+ self.add_fixed_orientation_mobjects(plane_y_x_2_text)
+ self.play(ShowCreation(limit_y_x_2))
+ self.wait()
+
+ self.play(ShowCreation(plane_y_2_x))
+ self.add_fixed_orientation_mobjects(plane_y_2_x_text)
+ self.play(ShowCreation(limit_y_2_x))
+ self.wait()
+
+ self.play(ShowCreation(line_1_1))
+ self.wait()
+
+ self.play(ShowCreation(point))
+ self.add_fixed_orientation_mobjects(point_text)
+ self.wait()
+
+ self.play(FadeOut(plane_y_x_text), FadeOut(plane_y_1_text), FadeOut(plane_y_x_2_text), FadeOut(plane_y_2_x_text))
+
+ self.move_camera(phi=0* DEGREES,theta=-95*DEGREES)
+ self.wait(2)
+ self.play(FadeOut(plane_y_x), FadeOut(plane_y_1), FadeOut(plane_y_x_2), FadeOut(plane_y_2_x))
+ self.wait(3)
+
+ self.move_camera(phi=75* DEGREES,theta=-95*DEGREES)
+ self.wait(3)
+
+
+
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file4_limit_different_point.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file4_limit_different_point.py new file mode 100644 index 0000000..0a43def --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file4_limit_different_point.py @@ -0,0 +1,115 @@ +from manimlib.imports import *
+
+class DifferentPoint(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ text3d = TextMobject(r"$f(x,y) = \frac{x^2 - y^2}{x^2 + y^2}$")
+ self.add_fixed_in_frame_mobjects(text3d)
+
+ text3d.to_corner(UL)
+
+ text3d.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+
+ self.play(Write(text3d))
+ self.wait(1)
+
+ limit_func = ParametricSurface(
+ lambda u, v: np.array([
+ 3*np.sin(u)*np.cos(v),
+ 3*np.sin(u)*np.sin(v),
+ (np.cos(v)*np.cos(v) - np.sin(v)*np.sin(v))
+ ]),u_min=0,u_max=PI,v_min=0,v_max=2*PI,checkerboard_colors=[YELLOW_C, YELLOW_E],
+ resolution=(15, 32)).scale(1)
+
+ limit_func_copy1 = limit_func.copy()
+ limit_func_copy2 = limit_func.copy()
+
+ limit_func_x = ParametricSurface(
+ lambda u, v: np.array([
+ 3*np.sin(u)*np.cos(v),
+ 3*np.sin(u)*np.sin(v),
+ (np.cos(v)*np.cos(v) - np.sin(v)*np.sin(v))
+ ]),u_min=0,u_max=PI,v_min=PI,v_max=2*PI,checkerboard_colors=[YELLOW_C, YELLOW_E],
+ resolution=(15, 32)).scale(1)
+
+ limit_func_y = ParametricSurface(
+ lambda u, v: np.array([
+ 3*np.sin(u)*np.cos(v),
+ 3*np.sin(u)*np.sin(v),
+ (np.cos(v)*np.cos(v) - np.sin(v)*np.sin(v))
+ ]),u_min=0,u_max=PI,v_min=PI/2,v_max=3*PI/2,checkerboard_colors=[YELLOW_C, YELLOW_E],
+ resolution=(15, 32)).scale(1)
+
+ limit_x =ParametricFunction(
+ lambda u : np.array([
+ u,
+ 0,
+ 1
+ ]),color="#006400",t_min=-3,t_max=3,
+ )
+
+ limit_y =ParametricFunction(
+ lambda u : np.array([
+ 0,
+ u,
+ -1
+ ]),color="#000080",t_min=-3,t_max=3,
+ )
+
+ plane_x = Polygon(np.array([-3,0,-2]),np.array([3,0,-2]),np.array([3,0,2]),np.array([-3,0,2]),np.array([-3,0,-2]), color = GREEN, fill_color = GREEN, fill_opacity = 0.2)
+ plane_x_text = TextMobject(r"$y = 0$", color = GREEN_C).move_to(1.7*UP + 3.8*RIGHT)
+
+ plane_y = Polygon(np.array([0,-3,-2]),np.array([0,3,-2]),np.array([0,3,2]),np.array([0,-3,2]),np.array([0,-3,-2]), color = BLUE, fill_color = BLUE, fill_opacity = 0.2)
+ plane_y_text = TextMobject(r"$x = 0$", color = BLUE_C).move_to(1.7*UP + 3.8*RIGHT)
+
+ origin_x = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).move_to(np.array([0,0,0]))
+ origin_x_text = TextMobject(r"$(0,0,0)$", color = RED_C).scale(0.7).move_to(np.array([-0.6,0,-0.5]))
+
+ origin_y = Polygon(*[np.array([0,0.05*np.cos(i*DEGREES),0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).move_to(np.array([0,0,0]))
+ origin_y_text = TextMobject(r"$(0,0,0)$", color = RED_C).scale(0.7).move_to(np.array([0,-0.6,-0.5]))
+
+ self.set_camera_orientation(phi=80 * DEGREES, theta = 0*DEGREES)
+
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(3.7*UP)
+
+ self.add_fixed_in_frame_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.play(ShowCreation(limit_func))
+
+ self.move_camera(phi=80* DEGREES,theta=105*DEGREES)
+
+ self.play(ShowCreation(plane_x))
+ self.add_fixed_in_frame_mobjects(plane_x_text)
+ self.wait()
+ self.play(ReplacementTransform(limit_func, limit_func_x))
+ self.play(FadeOut(plane_x), FadeOut(plane_x_text), ShowCreation(origin_x))
+ self.add_fixed_orientation_mobjects(origin_x_text)
+ self.play(ShowCreation(limit_x))
+
+ self.move_camera(phi=80* DEGREES,theta=15*DEGREES)
+ self.wait(3)
+
+ self.play(FadeOut(origin_x), FadeOut(origin_x_text), FadeOut(limit_x), ReplacementTransform(limit_func_x, limit_func_copy1))
+ self.play(ShowCreation(plane_y))
+ self.add_fixed_in_frame_mobjects(plane_y_text)
+ self.wait()
+ self.play(ReplacementTransform(limit_func_copy1, limit_func_y))
+ self.play(FadeOut(plane_y), FadeOut(plane_y_text), ShowCreation(origin_y))
+ self.add_fixed_orientation_mobjects(origin_y_text)
+ self.play(ShowCreation(limit_y))
+
+ self.move_camera(phi=80* DEGREES,theta=75*DEGREES)
+ self.wait(3)
+
+ self.play(FadeOut(origin_y), FadeOut(origin_y_text), FadeOut(limit_y), ReplacementTransform(limit_func_y, limit_func_copy2))
+ self.wait(2)
+
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file5_continuity_func.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file5_continuity_func.py new file mode 100644 index 0000000..99159a4 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file5_continuity_func.py @@ -0,0 +1,115 @@ +from manimlib.imports import *
+
+class Continuity(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ text3d = TextMobject(r"$f(x,y) = \frac{3x^2y}{x^2 + y^2}$")
+ self.add_fixed_in_frame_mobjects(text3d)
+
+ text3d.to_corner(UL)
+
+ text3d.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+
+ self.play(Write(text3d))
+ self.wait(1)
+
+
+ continuity_func = ParametricSurface(
+ lambda u, v: np.array([
+ 3*np.sin(u)*np.cos(v),
+ 3*np.sin(u)*np.sin(v),
+ 9*np.sin(u)*np.cos(v)*np.cos(v)*np.sin(v)
+ ]),u_min=0,u_max=PI,v_min=0,v_max=2*PI,checkerboard_colors=[YELLOW_C, YELLOW_E],
+ resolution=(15, 32)).scale(1)
+
+ continuity_func_copy1 = continuity_func.copy()
+ continuity_func_copy2 = continuity_func.copy()
+
+ continuity_func_x = ParametricSurface(
+ lambda u, v: np.array([
+ 3*np.sin(u)*np.cos(v),
+ 3*np.sin(u)*np.sin(v),
+ 9*np.sin(u)*np.cos(v)*np.cos(v)*np.sin(v)
+ ]),u_min=0,u_max=PI,v_min=PI,v_max=2*PI,checkerboard_colors=[YELLOW_C, YELLOW_E],
+ resolution=(15, 32)).scale(1)
+
+ continuity_func_y = ParametricSurface(
+ lambda u, v: np.array([
+ 3*np.sin(u)*np.cos(v),
+ 3*np.sin(u)*np.sin(v),
+ 9*np.sin(u)*np.cos(v)*np.cos(v)*np.sin(v)
+ ]),u_min=0,u_max=PI,v_min=PI/2,v_max=3*PI/2,checkerboard_colors=[YELLOW_C, YELLOW_E],
+ resolution=(15, 32)).scale(1)
+
+ continuity_x =ParametricFunction(
+ lambda u : np.array([
+ u,
+ 0,
+ 0
+ ]),color="#006400",t_min=-3,t_max=3,
+ )
+
+ continuity_y =ParametricFunction(
+ lambda u : np.array([
+ 0,
+ u,
+ 0
+ ]),color="#000080",t_min=-3,t_max=3,
+ )
+
+ plane_x = Polygon(np.array([-3,0,-3]),np.array([3,0,-3]),np.array([3,0,3]),np.array([-3,0,3]),np.array([-3,0,-3]), color = GREEN, fill_color = GREEN, fill_opacity = 0.2)
+ plane_x_text = TextMobject(r"$y = 0$", color = GREEN_C).move_to(1.7*UP + 3.8*RIGHT)
+
+ plane_y = Polygon(np.array([0,-3,-3]),np.array([0,3,-3]),np.array([0,3,3]),np.array([0,-3,3]),np.array([0,-3,-3]), color = BLUE, fill_color = BLUE, fill_opacity = 0.2)
+ plane_y_text = TextMobject(r"$x = 0$", color = BLUE_C).move_to(1.7*UP + 3.8*RIGHT)
+
+ origin_x = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).move_to(np.array([0,0,0]))
+ origin_x_text = TextMobject(r"$(0,0,0)$", color = RED_C).scale(0.7).move_to(np.array([-0.6,0,-0.5]))
+
+ origin_y = Polygon(*[np.array([0,0.05*np.cos(i*DEGREES),0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#006400", fill_color = "#006400", fill_opacity = 1).move_to(np.array([0,0,0]))
+ origin_y_text = TextMobject(r"$(0,0,0)$", color = RED_C).scale(0.7).move_to(np.array([0,-0.6,-0.5]))
+
+ self.set_camera_orientation(phi=80 * DEGREES, theta = 0*DEGREES)
+
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(3.7*UP)
+
+ self.add_fixed_in_frame_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.play(ShowCreation(continuity_func))
+
+ self.move_camera(phi=80* DEGREES,theta=105*DEGREES)
+
+ self.play(ShowCreation(plane_x))
+ self.add_fixed_in_frame_mobjects(plane_x_text)
+ self.wait()
+ self.play(ReplacementTransform(continuity_func, continuity_func_x))
+ self.play(FadeOut(plane_x), FadeOut(plane_x_text))
+ self.play(ShowCreation(continuity_x), ShowCreation(origin_x))
+ self.add_fixed_orientation_mobjects(origin_x_text)
+
+ self.move_camera(phi=80* DEGREES,theta=15*DEGREES)
+ self.wait(3)
+
+ self.play(FadeOut(origin_x), FadeOut(origin_x_text), FadeOut(continuity_x), ReplacementTransform(continuity_func_x, continuity_func_copy1))
+ self.play(ShowCreation(plane_y))
+ self.add_fixed_in_frame_mobjects(plane_y_text)
+ self.wait()
+ self.play(ReplacementTransform(continuity_func_copy1, continuity_func_y))
+ self.play(FadeOut(plane_y), FadeOut(plane_y_text))
+ self.play(ShowCreation(continuity_y), ShowCreation(origin_y))
+ self.add_fixed_orientation_mobjects(origin_y_text)
+
+ self.move_camera(phi=80* DEGREES,theta=75*DEGREES)
+ self.wait(3)
+
+ self.play(FadeOut(origin_y), FadeOut(origin_y_text), FadeOut(continuity_y), ReplacementTransform(continuity_func_y, continuity_func_copy2))
+ self.wait(2)
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file1_epsilon_delta_defn.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file1_epsilon_delta_defn.gif Binary files differnew file mode 100644 index 0000000..2378bcf --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file1_epsilon_delta_defn.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Limits and Continuity of Multivariable Functions/gifs/limit_approach_point.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file2_limit_approach_point.gif Binary files differindex 830b6f1..830b6f1 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Limits and Continuity of Multivariable Functions/gifs/limit_approach_point.gif +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file2_limit_approach_point.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file3_limit_approach_point_3d.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file3_limit_approach_point_3d.gif Binary files differnew file mode 100644 index 0000000..4bccf8c --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file3_limit_approach_point_3d.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file4_limit_different_point.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file4_limit_different_point.gif Binary files differnew file mode 100644 index 0000000..9a831e4 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file4_limit_different_point.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file5_continuity_func.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file5_continuity_func.gif Binary files differnew file mode 100644 index 0000000..2a0a61f --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file5_continuity_func.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file1_partial_deriv_gas_law.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file1_partial_deriv_gas_law.py new file mode 100644 index 0000000..3d35c97 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file1_partial_deriv_gas_law.py @@ -0,0 +1,88 @@ +from manimlib.imports import *
+
+class GasLaw(Scene):
+ def construct(self):
+ gas_law = TextMobject(r"$P$", r"$V$", r"=", r"$n$", r"$R$", r"$T$").scale(1.5)
+ gas_law[0].set_color(BLUE_C)
+ gas_law[1].set_color(GREEN_C)
+ gas_law[3].set_color(RED_C)
+ gas_law[4].set_color(ORANGE)
+ gas_law[5].set_color(YELLOW_C)
+
+ gas_law_trans = TexMobject("V", "=", "{n", "R", "T", "\\over", "P}").scale(1.5)
+ gas_law_trans[0].set_color(GREEN_C)
+ gas_law_trans[2].set_color(RED_C)
+ gas_law_trans[3].set_color(ORANGE)
+ gas_law_trans[4].set_color(YELLOW_C)
+ gas_law_trans[6].set_color(BLUE_C)
+
+ gas_law_func = TexMobject("V", "=", "f(", "n", ",", "T", ",", "P", ")").scale(1.5)
+ gas_law_func[0].set_color(GREEN_C)
+ gas_law_func[2].set_color(ORANGE)
+ gas_law_func[3].set_color(RED_C)
+ gas_law_func[5].set_color(YELLOW_C)
+ gas_law_func[7].set_color(BLUE_C)
+ gas_law_func[8].set_color(ORANGE)
+
+ partial_gas_law_func = TexMobject("{\\partial", "V","\\over", "\\partial", "P}", r"=", "{\\partial", "\\over", "\\partial", "P}", "f(", r"n", ",", r"T", ",", r"P", r")").scale(1.5)
+ partial_gas_law_func.set_color_by_tex("\\partial", PINK)
+ partial_gas_law_func.set_color_by_tex("P}", BLUE_C)
+
+ partial_gas_law_func[1].set_color(GREEN_C)
+ partial_gas_law_func[10].set_color(ORANGE)
+ partial_gas_law_func[11].set_color(RED_C)
+ partial_gas_law_func[13].set_color(YELLOW_C)
+ partial_gas_law_func[15].set_color(BLUE_C)
+ partial_gas_law_func[16].set_color(ORANGE)
+
+ partial_gas_law_trans = TexMobject("{\\partial", "V","\\over", "\\partial", "P}", r"=", "{\\partial", "\\over", "\\partial", "P}", "{n", "R", "T", "\\over", "P}").scale(1.5)
+ partial_gas_law_trans.set_color_by_tex("\\partial", PINK)
+ partial_gas_law_trans.set_color_by_tex("P}", BLUE_C)
+
+ partial_gas_law_trans[1].set_color(GREEN_C)
+ partial_gas_law_trans[10].set_color(RED_C)
+ partial_gas_law_trans[11].set_color(ORANGE)
+ partial_gas_law_trans[12].set_color(YELLOW_C)
+
+ partial_gas_law_trans2 = TexMobject("{\\partial", "V","\\over", "\\partial", "P}", r"=", "n", "R", "T", "{\\partial", "\\over", "\\partial", "P}", "P^{-1}",).scale(1.5)
+ partial_gas_law_trans2.set_color_by_tex("\\partial", PINK)
+ partial_gas_law_trans2.set_color_by_tex("P}", BLUE_C)
+
+ partial_gas_law_trans2[1].set_color(GREEN_C)
+ partial_gas_law_trans2[6].set_color(RED_C)
+ partial_gas_law_trans2[7].set_color(ORANGE)
+ partial_gas_law_trans2[8].set_color(YELLOW_C)
+ partial_gas_law_trans2[-1].set_color(BLUE_C)
+
+ partial_gas_law_trans3 = TexMobject("{\\partial", "V","\\over", "\\partial", "P}", r"=", "n", "R", "T", "P^{-2}",).scale(1.5)
+ partial_gas_law_trans3.set_color_by_tex("\\partial", PINK)
+ partial_gas_law_trans3.set_color_by_tex("P}", BLUE_C)
+
+ partial_gas_law_trans3[1].set_color(GREEN_C)
+ partial_gas_law_trans3[6].set_color(RED_C)
+ partial_gas_law_trans3[7].set_color(ORANGE)
+ partial_gas_law_trans3[8].set_color(YELLOW_C)
+ partial_gas_law_trans3[9].set_color(BLUE_C)
+
+ framebox = SurroundingRectangle(partial_gas_law_trans3, color = PURPLE, buff = 0.3)
+
+
+
+ self.play(Write(gas_law))
+ self.wait()
+ self.play(Transform(gas_law, gas_law_trans))
+ self.wait()
+ self.play(Transform(gas_law, gas_law_func))
+ self.wait()
+ self.play(Transform(gas_law, gas_law_trans))
+ self.wait()
+ self.play(Transform(gas_law, partial_gas_law_func))
+ self.wait()
+ self.play(Transform(gas_law, partial_gas_law_trans))
+ self.wait()
+ self.play(Transform(gas_law, partial_gas_law_trans2))
+ self.wait()
+ self.play(Transform(gas_law, partial_gas_law_trans3))
+ self.wait()
+ self.play(ShowCreation(framebox))
+ self.wait()
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file2_partial_deriv_hill.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file2_partial_deriv_hill.py new file mode 100644 index 0000000..bfb7687 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file2_partial_deriv_hill.py @@ -0,0 +1,122 @@ +from manimlib.imports import *
+
+class Hill(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ function = ParametricSurface(
+ lambda u, v: np.array([
+ 1.2*np.sin(u)*np.cos(v),
+ 1.2*np.sin(u)*np.sin(v),
+ -1.2*1.2*np.sin(u)*np.sin(u)*(1+0.5*np.sin(v)*np.sin(v))+2
+ ]),u_min=0,u_max=PI/2,v_min=0,v_max=2*PI,checkerboard_colors=[GREEN_C, GREEN_E],
+ resolution=(15, 32)).scale(1)
+
+ func_x =ParametricFunction(
+ lambda u : np.array([
+ u,
+ 0,
+ 2 - u*u
+ ]),color=RED_E,t_min=-1.2,t_max=1.2,
+ )
+
+ func_y =ParametricFunction(
+ lambda u : np.array([
+ 0,
+ u,
+ 2 - 1.5*u*u
+ ]),color=PINK,t_min=-1.2,t_max=1.2,
+ )
+
+ self.set_camera_orientation(phi=60 * DEGREES, theta = 0*DEGREES)
+ #self.set_camera_orientation(phi=45 * DEGREES, theta = -20*DEGREES)
+
+ self.add(axes)
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.play(ShowCreation(function))
+ self.wait()
+
+ self.move_camera(phi=60 * DEGREES, theta = 45*DEGREES)
+ #self.play(ShowCreation(func_x))
+
+ text_x = TextMobject("Slope of the hill along", r"$x$", "axis", color = YELLOW_C).scale(0.6).move_to(2.7*UP + 3.5*RIGHT)
+ text_x[1].set_color(PINK)
+
+
+ slope_text_x = TexMobject("Slope =", "{\\partial", "f", "\\over", "\\partial", "x}").scale(0.6).move_to(2*UP + 3.5*RIGHT)
+ slope_text_x[0].set_color(BLUE_E)
+ slope_text_x.set_color_by_tex("\\partial",YELLOW_C)
+ slope_text_x.set_color_by_tex("f",RED_E)
+ slope_text_x[5].set_color(PINK)
+
+ self.add_fixed_in_frame_mobjects(text_x, slope_text_x)
+
+ dot_x = Dot().rotate(PI/2).set_color(YELLOW_E)
+ alpha_x = ValueTracker(0)
+ vector_x = self.get_tangent_vector(alpha_x.get_value(),func_x,scale=1.5)
+ dot_x.add_updater(lambda m: m.move_to(vector_x.get_center()))
+ self.play(
+ ShowCreation(func_x),
+ GrowFromCenter(dot_x),
+ GrowArrow(vector_x)
+ )
+ vector_x.add_updater(
+ lambda m: m.become(
+ self.get_tangent_vector(alpha_x.get_value()%1,func_x,scale=1.5)
+ )
+ )
+
+ self.add(vector_x,dot_x)
+
+ self.play(alpha_x.increment_value, 1, run_time=10, rate_func=linear)
+
+ #self.move_camera(phi=60 * DEGREES, theta = 0*DEGREES)
+ self.play(FadeOut(vector_x), FadeOut(dot_x), FadeOut(func_x), FadeOut(text_x), FadeOut(slope_text_x))
+
+ text_y = TextMobject("Slope of the hill along", r"$y$", "axis", color = YELLOW_C).scale(0.6).move_to(2.7*UP + 3.5*RIGHT)
+ text_y[1].set_color(RED_C)
+
+
+ slope_text_y = TexMobject("Slope =", "{\\partial", "f", "\\over", "\\partial", "x}").scale(0.6).move_to(2*UP + 3.5*RIGHT)
+ slope_text_y[0].set_color(BLUE_E)
+ slope_text_y.set_color_by_tex("\\partial",YELLOW_C)
+ slope_text_y.set_color_by_tex("f",PINK)
+ slope_text_y[5].set_color(RED_C)
+
+ self.add_fixed_in_frame_mobjects(text_y, slope_text_y)
+
+ dot_y = Dot().rotate(PI/2).set_color(BLUE_E)
+ alpha_y = ValueTracker(0)
+ vector_y = self.get_tangent_vector(alpha_y.get_value(),func_y,scale=1.5)
+ dot_y.add_updater(lambda m: m.move_to(vector_y.get_center()))
+ self.play(
+ ShowCreation(func_y),
+ GrowFromCenter(dot_y),
+ GrowArrow(vector_y)
+ )
+ vector_y.add_updater(
+ lambda m: m.become(
+ self.get_tangent_vector(alpha_y.get_value()%1,func_y,scale=1.5)
+ )
+ )
+
+ self.add(vector_y,dot_y)
+ self.play(alpha_y.increment_value, 1, run_time=10, rate_func=linear)
+ self.play(FadeOut(vector_y), FadeOut(dot_y), FadeOut(func_y), FadeOut(text_y), FadeOut(slope_text_y))
+ self.wait(2)
+
+ def get_tangent_vector(self, proportion, curve, dx=0.001, scale=1):
+ coord_i = curve.point_from_proportion(proportion)
+ coord_f = curve.point_from_proportion(proportion + dx)
+ reference_line = Line(coord_i,coord_f)
+ unit_vector = reference_line.get_unit_vector() * scale
+ vector = Line(coord_i - unit_vector, coord_i + unit_vector, color = ORANGE, buff=0)
+ return vector
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file3_partial_deriv_defn.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file3_partial_deriv_defn.py new file mode 100644 index 0000000..a25ca56 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file3_partial_deriv_defn.py @@ -0,0 +1,218 @@ +from manimlib.imports import *
+
+class PartialDeriv(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ paraboloid = ParametricSurface(
+ lambda u, v: np.array([
+ 2*np.sin(u)*np.cos(v),
+ 2*np.sin(u)*np.sin(v),
+ -2*2*np.sin(u)*np.sin(u)+2
+ ]),u_min=0,u_max=PI/2,v_min=0,v_max=2*PI,checkerboard_colors=[PINK, PURPLE],
+ resolution=(15, 32)).scale(1)
+
+ paraboloid_copy1 = paraboloid.copy()
+ paraboloid_copy2 = paraboloid.copy()
+
+ paraboloid_x = ParametricSurface(
+ lambda u, v: np.array([
+ 2*np.sin(u)*np.cos(v),
+ 2*np.sin(u)*np.sin(v),
+ -2*2*np.sin(u)*np.sin(u)+2
+ ]),u_min=0,u_max=PI/2,v_min=PI,v_max=2*PI,checkerboard_colors=[PINK, PURPLE],
+ resolution=(15, 32)).scale(1)
+
+ paraboloid_x_copy = paraboloid_x.copy()
+
+ paraboloid_y = ParametricSurface(
+ lambda u, v: np.array([
+ 2*np.sin(u)*np.cos(v),
+ 2*np.sin(u)*np.sin(v),
+ -2*2*np.sin(u)*np.sin(u)+2
+ ]),u_min=0,u_max=PI/2,v_min=PI/2,v_max=3*PI/2,checkerboard_colors=[PINK, PURPLE],
+ resolution=(15, 32)).scale(1)
+
+ parabola1 =ParametricFunction(
+ lambda u : np.array([
+ u,
+ 0,
+ -(u*u) + 2
+ ]),color="#006400",t_min=-2,t_max=2,
+ )
+ parabola2 =ParametricFunction(
+ lambda u : np.array([
+ 0,
+ u,
+ -(u*u) + 2
+ ]),color=BLUE_C,t_min=-2,t_max=2,
+ )
+
+ plane1 = Polygon(np.array([-2.2,0,-2.5]),np.array([2.2,0,-2.5]),np.array([2.2,0,2.5]),np.array([-2.2,0,2.5]),np.array([-2.2,0,-2.5]), color = GREEN, fill_color = GREEN, fill_opacity = 0.2)
+ plane1_text = TextMobject(r"$y = 0$", color = GREEN_C).move_to(2*UP + 3.3*RIGHT)
+
+ plane2 = Polygon(np.array([0,-2.2,-2.5]),np.array([0,2.2,-2.5]),np.array([0,2.2,2.5]),np.array([0,-2.2,2.5]),np.array([0,-2.2,-2.5]), color = BLUE, fill_color = BLUE, fill_opacity = 0.2)
+ plane2_text = TextMobject(r"$x = 0$", color = BLUE_C).move_to(2*UP + 3.2*RIGHT)
+
+ surface_eqn = TextMobject("Surface", r"$z = 2- x^2 -y^2$", color = YELLOW_C).scale(0.6).move_to(np.array([3*LEFT +3*UP]))
+ surface_eqn[0].set_color(PINK)
+
+ dot1 =Sphere(radius=0.08).move_to(np.array([-1,0,1]))
+ dot1.set_fill(RED)
+ line1 = Line(np.array([-1.55, 0,0]), np.array([-0.4, 0,2.2]), color = RED)
+ lab_x = TextMobject(r"$f(x_0,y_0)$", color = RED).scale(0.7)
+ para_lab_x = TextMobject(r"$f(x,y_0)$", color = "#006400").scale(0.7)
+ tangent_line_x = TextMobject("Tangent Line", color = RED_C, buff = 0.4).scale(0.6).move_to(np.array([1.7*RIGHT +1.8*UP]))
+
+
+ text1 = TextMobject(r"$\frac{\partial f}{\partial x}\vert_{(x_0,y_0)} = \frac{d}{dx}$", r"$f(x,y_0)$", r"$\vert_{x=x_0}$").scale(0.6)
+ brace1 = Brace(text1[1], DOWN, buff = SMALL_BUFF, color = GREEN)
+ t1 = brace1.get_text("Just depends on x")
+ t1.scale(0.6)
+ t1.set_color(GREEN)
+
+
+ dot2 =Sphere(radius=0.08).move_to(np.array([0,1,1]))
+ dot2.set_fill(RED)
+ line2 = Line(np.array([0, 1.55,0]), np.array([0, 0.4,2.2]), color = RED)
+ lab_y = TextMobject(r"$f(x_0,y_0)$", color = RED).scale(0.7)
+ para_lab_y = TextMobject(r"$f(x_0,y)$", color = BLUE_C).scale(0.7)
+ tangent_line_y = TextMobject("Tangent Line", color = RED_C, buff = 0.4).scale(0.6).move_to(np.array([1.7*RIGHT +1.8*UP]))
+
+ text2 = TextMobject(r"$\frac{\partial f}{\partial y}\vert_{(x_0,y_0)} = \frac{d}{dy}$", r"$f(x_0,y)$", r"$\vert_{y=y_0}$").scale(0.6)
+ brace2 = Brace(text2[1], DOWN, buff = SMALL_BUFF, color = GREEN)
+ t2 = brace2.get_text("Just depends on y")
+ t2.scale(0.6)
+ t2.set_color(GREEN)
+
+ text3 = TextMobject(r"$= \lim_{h \to 0} \frac{f(x_0+h,y_0) - f(x_0,y_0)}{h}$").scale(0.6)
+
+ dot3 =Sphere(radius=0.08).move_to(np.array([-1.22,0,0.5]))
+ dot3.set_fill(YELLOW_C)
+ line3 = Line(np.array([-1.44,0,0]), np.array([-0.6,0,2.2]), color = YELLOW_C)
+ lab_line3 = TextMobject(r"$f(x_0+h,y_0)$", color = YELLOW_C).scale(0.7)
+
+
+ self.set_camera_orientation(phi=80 * DEGREES, theta = 0*DEGREES)
+ #self.set_camera_orientation(phi=80 * DEGREES, theta = 20*DEGREES)
+ #self.begin_ambient_camera_rotation(rate=0.3)
+
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(3.7*UP)
+
+ self.add_fixed_in_frame_mobjects(axis[2])
+ #self.add_fixed_orientation_mobjects(axis[2])
+
+ self.play(Write(paraboloid))
+
+ self.add_fixed_in_frame_mobjects(surface_eqn)
+ #self.move_camera(phi=80* DEGREES,theta=110*DEGREES)
+ self.move_camera(phi=80* DEGREES,theta=45*DEGREES)
+
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+ self.play(ShowCreation(plane1))
+ self.add_fixed_in_frame_mobjects(plane1_text)
+ self.wait()
+ self.play(ReplacementTransform(paraboloid, paraboloid_x))
+
+ lab_x.move_to(np.array([1.8*RIGHT +1.15*UP]))
+ para_lab_x.move_to(np.array([1.3*LEFT +1.6*UP]))
+ self.wait()
+ self.play(FadeOut(plane1), FadeOut(plane1_text))
+ self.play(ShowCreation(parabola1))
+ self.add_fixed_in_frame_mobjects(para_lab_x)
+ self.play(ShowCreation(dot1))
+ self.add_fixed_in_frame_mobjects(lab_x)
+ #self.play(ShowCreation(dot1))
+ self.wait()
+ self.play(ShowCreation(line1))
+ self.add_fixed_in_frame_mobjects(tangent_line_x)
+ self.wait()
+
+ self.add_fixed_in_frame_mobjects(text1, brace1, t1)
+ grp1 = VGroup(text1, brace1, t1)
+ grp1.move_to(3*UP+3*RIGHT)
+ self.play(Write(text1),GrowFromCenter(brace1), FadeIn(t1))
+ self.wait()
+ self.play(FadeOut(parabola1), FadeOut(line1), FadeOut(lab_x), FadeOut(para_lab_x), FadeOut(dot1), FadeOut(tangent_line_x),FadeOut(grp1))
+
+
+
+
+ #self.move_camera(phi=80* DEGREES,theta=20*DEGREES)
+
+ self.play(ReplacementTransform(paraboloid_x, paraboloid_copy1))
+ self.wait()
+ self.play(ShowCreation(plane2))
+ self.add_fixed_in_frame_mobjects(plane2_text)
+ self.wait()
+ self.play(ReplacementTransform(paraboloid_copy1, paraboloid_y))
+
+ lab_y.move_to(np.array([1.8*RIGHT +1.15*UP]))
+ para_lab_y.move_to(np.array([1.3*LEFT +1.6*UP]))
+ self.wait()
+ self.play(FadeOut(plane2), FadeOut(plane2_text))
+ self.play(ShowCreation(parabola2))
+ self.add_fixed_in_frame_mobjects(para_lab_y)
+ self.play(ShowCreation(dot2))
+ self.add_fixed_in_frame_mobjects(lab_y)
+ self.wait()
+ self.play(ShowCreation(line2))
+ self.add_fixed_in_frame_mobjects(tangent_line_y)
+ self.wait()
+
+ self.add_fixed_in_frame_mobjects(text2, brace2, t2)
+ grp2 = VGroup(text2, brace2, t2)
+ grp2.move_to(3*UP+3*RIGHT)
+ self.play(Write(text2),GrowFromCenter(brace2), FadeIn(t2))
+ self.wait()
+ self.play(FadeOut(parabola2), FadeOut(line2), FadeOut(lab_y), FadeOut(para_lab_y), FadeOut(dot2), FadeOut(tangent_line_y), FadeOut(grp2))
+ self.wait()
+
+
+ #self.move_camera(phi=80* DEGREES,theta=105*DEGREES)
+ self.play(ReplacementTransform(paraboloid_y, paraboloid_copy2))
+ self.wait()
+
+
+ self.play(ShowCreation(plane1))
+ self.add_fixed_in_frame_mobjects(plane1_text)
+ self.wait()
+ self.play(ReplacementTransform(paraboloid_copy2, paraboloid_x_copy))
+
+ lab_x.move_to(np.array([1.8*RIGHT +1.15*UP]))
+ para_lab_x.move_to(np.array([1.3*LEFT +1.6*UP]))
+ lab_line3.move_to(np.array([2.4*RIGHT +0.5*UP]))
+ self.wait()
+ self.play(FadeOut(plane1), FadeOut(plane1_text))
+ self.play(ShowCreation(parabola1))
+ self.add_fixed_in_frame_mobjects(para_lab_x)
+ self.play(ShowCreation(dot1))
+ self.add_fixed_in_frame_mobjects(lab_x)
+ self.play(ShowCreation(dot3))
+ self.add_fixed_in_frame_mobjects(lab_line3)
+ self.wait()
+ self.play(ShowCreation(line1))
+ self.add_fixed_in_frame_mobjects(tangent_line_x)
+ self.play(ShowCreation(line3))
+ self.wait()
+
+
+ self.add_fixed_in_frame_mobjects(text1,text3)
+ text1.move_to(3*UP+3*RIGHT)
+ text3.next_to(text1, DOWN)
+ self.play(Write(text1),Write(text3))
+ self.wait()
+ self.play(FadeOut(parabola1), FadeOut(line1), FadeOut(lab_x), FadeOut(line3), FadeOut(lab_line3), FadeOut(para_lab_x), FadeOut(dot1), FadeOut(dot3), FadeOut(tangent_line_x), FadeOut(text1), FadeOut(text3))
+ self.wait()
+
+
+
+
+
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file4_partial_deriv_example.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file4_partial_deriv_example.py new file mode 100644 index 0000000..5712a62 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file4_partial_deriv_example.py @@ -0,0 +1,246 @@ +from manimlib.imports import *
+
+class PartialDerivX(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ paraboloid = ParametricSurface(
+ lambda u, v: np.array([
+ 2*np.sin(u)*np.cos(v),
+ 2*np.sin(u)*np.sin(v),
+ -2*2*np.sin(u)*np.sin(u)+2
+ ]),u_min=0,u_max=PI/2,v_min=0,v_max=2*PI,checkerboard_colors=[PINK, PURPLE],
+ resolution=(15, 32)).scale(1)
+
+ paraboloid_copy = paraboloid.copy()
+
+
+ paraboloid_x = ParametricSurface(
+ lambda u, v: np.array([
+ 2*np.sin(u)*np.cos(v),
+ 2*np.sin(u)*np.sin(v),
+ -2*2*np.sin(u)*np.sin(u)+2
+ ]),u_min=0,u_max=PI/2,v_min=PI,v_max=2*PI,checkerboard_colors=[PINK, PURPLE],
+ resolution=(15, 32)).scale(1)
+
+
+ parabola =ParametricFunction(
+ lambda u : np.array([
+ u,
+ 0,
+ -(u*u) + 2
+ ]),color="#006400",t_min=-2,t_max=2,
+ )
+
+ plane = Polygon(np.array([-2.2,0,-2.5]),np.array([2.2,0,-2.5]),np.array([2.2,0,2.5]),np.array([-2.2,0,2.5]),np.array([-2.2,0,-2.5]), color = GREEN, fill_color = GREEN, fill_opacity = 0.2)
+ plane_text = TextMobject(r"$y = 0$", color = GREEN_C).move_to(2*UP + 3*RIGHT)
+
+ surface_eqn = TextMobject("Surface", r"$z = 2- x^2 -y^2$", color = PINK).scale(0.6).move_to(np.array([3*LEFT +3*UP]))
+ surface_eqn[0].set_color(BLUE_C)
+
+ line = Line(np.array([-2,0,0]), np.array([2,0,0]), color = RED_C)
+
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(3.7*UP)
+
+ self.add_fixed_in_frame_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+
+ self.set_camera_orientation(phi=80 * DEGREES, theta = 0*DEGREES)
+
+ self.play(Write(paraboloid))
+
+ self.add_fixed_in_frame_mobjects(surface_eqn)
+ #self.move_camera(phi=80* DEGREES,theta=95*DEGREES)
+ self.move_camera(phi=80* DEGREES,theta=45*DEGREES)
+ self.play(ShowCreation(plane))
+ self.add_fixed_in_frame_mobjects(plane_text)
+ self.wait()
+ self.play(ReplacementTransform(paraboloid, paraboloid_x))
+ self.play(FadeOut(plane), FadeOut(plane_text))
+ self.play(ShowCreation(parabola), ShowCreation(line))
+
+ text1 = TextMobject("Moving small", r"$dx$", r"steps").scale(0.6).move_to(3*UP + 3.5*RIGHT).set_color_by_gradient(RED, ORANGE, YELLOW, BLUE, PURPLE)
+
+ text2 = TextMobject("Observing change in function, keeping", r"$y$", r"constant").scale(0.6).move_to(2.6*UP + 3.5*RIGHT).set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+
+ slope_text = TexMobject("Slope =", "{\\partial", "f", "\\over", "\\partial", "x}").scale(0.6).move_to(2*UP + 3.5*RIGHT)
+ slope_text[0].set_color(BLUE_E)
+ slope_text.set_color_by_tex("\\partial",PINK)
+ slope_text.set_color_by_tex("f","#006400")
+ slope_text[5].set_color(RED_C)
+
+ self.add_fixed_in_frame_mobjects(text1, text2)
+ self.wait()
+ self.add_fixed_in_frame_mobjects(slope_text)
+ #add_fixed_orientation_mobjects
+
+
+ dot = Dot().rotate(PI/2).set_color(RED_C)
+ alpha = ValueTracker(0)
+ vector = self.get_tangent_vector(alpha.get_value(),parabola,scale=1.5)
+ dot.add_updater(lambda m: m.move_to(vector.get_center()))
+ self.play(
+ ShowCreation(parabola),
+ GrowFromCenter(dot),
+ GrowArrow(vector)
+ )
+ vector.add_updater(
+ lambda m: m.become(
+ self.get_tangent_vector(alpha.get_value()%1,parabola,scale=1.5)
+ )
+ )
+ self.add(vector,dot)
+ self.play(alpha.increment_value, 1, run_time=10, rate_func=linear)
+ self.wait()
+
+
+ '''
+ for i in np.arange(-2,2,0.2):
+ self.play(ReplacementTransform(Line(np.array([i,0,0]), np.array([i,0,-i*i + 2]), color = GREEN_C), Line(np.array([i+0.2,0,0]), np.array([i+0.2,0,-(i+0.2)**2 + 2]), color = GREEN_C)))
+ #self.wait()
+ '''
+
+ self.wait()
+ self.play(FadeOut(parabola), FadeOut(line), FadeOut(vector), FadeOut(dot), FadeOut(text1), FadeOut(text2), FadeOut(slope_text),FadeOut(surface_eqn))
+
+ #self.move_camera(phi=80* DEGREES,theta= 0*DEGREES)
+ self.play(ReplacementTransform(paraboloid_x, paraboloid_copy))
+ self.wait()
+
+
+ def get_tangent_vector(self, proportion, curve, dx=0.001, scale=1):
+ coord_i = curve.point_from_proportion(proportion)
+ coord_f = curve.point_from_proportion(proportion + dx)
+ reference_line = Line(coord_i,coord_f)
+ unit_vector = reference_line.get_unit_vector() * scale
+ vector = Line(coord_i - unit_vector, coord_i + unit_vector, color = BLUE_E, buff=0)
+ return vector
+
+
+class PartialDerivY(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ paraboloid = ParametricSurface(
+ lambda u, v: np.array([
+ 2*np.sin(u)*np.cos(v),
+ 2*np.sin(u)*np.sin(v),
+ -2*2*np.sin(u)*np.sin(u)+2
+ ]),u_min=0,u_max=PI/2,v_min=0,v_max=2*PI,checkerboard_colors=[PINK, PURPLE],
+ resolution=(15, 32)).scale(1)
+
+ paraboloid_copy = paraboloid.copy()
+
+
+ paraboloid_y = ParametricSurface(
+ lambda u, v: np.array([
+ 2*np.sin(u)*np.cos(v),
+ 2*np.sin(u)*np.sin(v),
+ -2*2*np.sin(u)*np.sin(u)+2
+ ]),u_min=0,u_max=PI/2,v_min=PI/2,v_max=3*PI/2,checkerboard_colors=[PINK, PURPLE],
+ resolution=(15, 32)).scale(1)
+
+
+ parabola =ParametricFunction(
+ lambda u : np.array([
+ 0,
+ u,
+ -(u*u) + 2
+ ]),color=YELLOW_C,t_min=-2,t_max=2,
+ )
+
+ plane = Polygon(np.array([0,-2.2,-2.5]),np.array([0,2.2,-2.5]),np.array([0,2.2,2.5]),np.array([0,-2.2,2.5]),np.array([0,-2.2,-2.5]), color = BLUE, fill_color = BLUE, fill_opacity = 0.2)
+ plane_text = TextMobject(r"$x = 0$", color = BLUE_C).move_to(2*UP + 3*RIGHT)
+
+ surface_eqn = TextMobject("Surface", r"$z = 2- x^2 -y^2$", color = PINK).scale(0.6).move_to(np.array([3*LEFT +3*UP]))
+ surface_eqn[0].set_color(BLUE_C)
+
+ line = Line(np.array([0,-2,0]), np.array([0,2,0]), color = RED_C)
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(3.7*UP)
+
+ self.add_fixed_in_frame_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.set_camera_orientation(phi=80 * DEGREES, theta = 45*DEGREES)
+
+ self.play(Write(paraboloid))
+
+ self.add_fixed_in_frame_mobjects(surface_eqn)
+ #self.move_camera(phi=80* DEGREES,theta=5*DEGREES)
+ self.play(ShowCreation(plane))
+ self.add_fixed_in_frame_mobjects(plane_text)
+ self.wait()
+ self.play(ReplacementTransform(paraboloid, paraboloid_y))
+ self.play(FadeOut(plane), FadeOut(plane_text))
+ self.play(ShowCreation(parabola), ShowCreation(line))
+
+ text1 = TextMobject("Moving small", r"$dy$", r"steps").scale(0.6).move_to(3*UP + 3.5*RIGHT).set_color_by_gradient(RED, ORANGE, YELLOW, BLUE, PURPLE)
+
+ text2 = TextMobject("Observing change in function, keeping", r"$x$", r"constant").scale(0.6).move_to(2.6*UP + 3.5*RIGHT).set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+
+ slope_text = TexMobject("Slope =", "{\\partial", "f", "\\over", "\\partial", "y}").scale(0.6).move_to(2*UP + 3.5*RIGHT)
+ slope_text[0].set_color("#006400")
+ slope_text.set_color_by_tex("\\partial",PINK)
+ slope_text.set_color_by_tex("f",YELLOW_C)
+ slope_text[5].set_color(RED_C)
+
+ self.add_fixed_in_frame_mobjects(text1, text2)
+ self.wait()
+ self.add_fixed_in_frame_mobjects(slope_text)
+
+ dot = Dot().rotate(PI/2).set_color(RED_C)
+ alpha = ValueTracker(0)
+ vector = self.get_tangent_vector(alpha.get_value(),parabola,scale=1.5)
+ dot.add_updater(lambda m: m.move_to(vector.get_center()))
+ self.play(
+ ShowCreation(parabola),
+ GrowFromCenter(dot),
+ GrowArrow(vector)
+ )
+ vector.add_updater(
+ lambda m: m.become(
+ self.get_tangent_vector(alpha.get_value()%1,parabola,scale=1.5)
+ )
+ )
+ self.add(vector,dot)
+ self.play(alpha.increment_value, 1, run_time=10, rate_func=linear)
+ self.wait()
+
+ '''
+ for i in np.arange(-2,2,0.2):
+ self.play(ReplacementTransform(Line(np.array([0,i,0]), np.array([0,i,-i*i + 2]), color = BLUE_C), Line(np.array([0,i+0.2,0]), np.array([0,i+0.2,-(i+0.2)**2 + 2]), color = BLUE_C)))
+ #self.wait()
+ '''
+
+
+ self.wait()
+ self.play(FadeOut(parabola), FadeOut(line), FadeOut(vector), FadeOut(dot), FadeOut(text1), FadeOut(text2), FadeOut(slope_text),FadeOut(surface_eqn))
+
+ #self.move_camera(phi=80* DEGREES,theta= 90*DEGREES)
+ self.play(ReplacementTransform(paraboloid_y, paraboloid_copy))
+ self.wait()
+
+ def get_tangent_vector(self, proportion, curve, dx=0.001, scale=1):
+ coord_i = curve.point_from_proportion(proportion)
+ coord_f = curve.point_from_proportion(proportion + dx)
+ reference_line = Line(coord_i,coord_f)
+ unit_vector = reference_line.get_unit_vector() * scale
+ vector = Line(coord_i - unit_vector, coord_i + unit_vector, color = "#006400", buff=0)
+ return vector
+
+
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file5_partial_deriv_func_2maximas.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file5_partial_deriv_func_2maximas.py new file mode 100644 index 0000000..7bbb9a7 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file5_partial_deriv_func_2maximas.py @@ -0,0 +1,227 @@ +from manimlib.imports import *
+
+class MaximaMinima(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ paraboloid = ParametricSurface(
+ lambda u, v: np.array([
+ 3.5*np.sin(u)*np.cos(v),
+ 3.5*np.sin(u)*np.sin(v),
+ 3.5*3.5*np.sin(u)*np.sin(u)*(1+2*np.sin(v)*np.sin(v))*np.exp(1 - 3.5*3.5*np.sin(u)*np.sin(u) )
+ ]),u_min=0,u_max=PI,v_min=0,v_max=2*PI, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1,
+ resolution=(15, 32)).scale(1)
+
+ paraboloid_copy1 = paraboloid.copy()
+ paraboloid_copy2 = paraboloid.copy()
+
+ paraboloid_x = ParametricSurface(
+ lambda u, v: np.array([
+ 3.5*np.sin(u)*np.cos(v),
+ 3.5*np.sin(u)*np.sin(v),
+ 3.5*3.5*np.sin(u)*np.sin(u)*(1+2*np.sin(v)*np.sin(v))*np.exp(1 - 3.5*3.5*np.sin(u)*np.sin(u) )
+ ]),u_min=0,u_max=PI,v_min=PI,v_max=2*PI, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1,
+ resolution=(15, 32)).scale(1)
+
+ paraboloid_y = ParametricSurface(
+ lambda u, v: np.array([
+ 3.5*np.sin(u)*np.cos(v),
+ 3.5*np.sin(u)*np.sin(v),
+ 3.5*3.5*np.sin(u)*np.sin(u)*(1+2*np.sin(v)*np.sin(v))*np.exp(1 - 3.5*3.5*np.sin(u)*np.sin(u) )
+ ]),u_min=0,u_max=PI,v_min=PI/2,v_max=3*PI/2, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1,
+ resolution=(15, 32)).scale(1)
+
+ parabola_x_out =ParametricFunction(
+ lambda u : np.array([
+ u,
+ 0,
+ (u*u )*np.exp(1-u*u)
+ ]),color=RED_E,t_min=-3.5,t_max=3.5,
+ )
+
+ parabola_y_out =ParametricFunction(
+ lambda u : np.array([
+ 0,
+ u,
+ (3*u*u)*np.exp(1-u*u)
+ ]),color=PINK,t_min=-3.5,t_max=3.5,
+ )
+
+ plane1 = Polygon(np.array([-3.5,0,-3]),np.array([3.5,0,-3]),np.array([3.5,0,3]),np.array([-3.5,0,3]),np.array([-3.5,0,-3]), color = RED_C, fill_color = RED_C, fill_opacity = 0.2)
+ plane_text_x = TextMobject(r"$y = 0$", color = RED_C).move_to(2*UP + 4.5*RIGHT)
+
+ plane2 = Polygon(np.array([0,-3.5,-3]),np.array([0,3.5,-3]),np.array([0,3.5,3]),np.array([0,-3.5,3]),np.array([0,-3.5,-3]), color = PINK, fill_color = PINK, fill_opacity = 0.2)
+ plane_text_y = TextMobject(r"$x = 0$", color = PINK).move_to(2*UP + 4.5*RIGHT)
+
+ surface_eqn = TextMobject("Surface", r"$z = (x^2 + 3y^2)e^{(1 - x^2 - y^2)}$", color = YELLOW_C).scale(0.6).move_to(np.array([3.5*LEFT +3.5*UP]))
+ surface_eqn[0].set_color(BLUE_C)
+
+ self.set_camera_orientation(phi=60 * DEGREES, theta = 45*DEGREES)
+
+ self.add(axes)
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.play(ShowCreation(paraboloid))
+
+
+ #self.move_camera(phi=60 * DEGREES, theta = 45*DEGREES,run_time=3)
+
+
+ plane_x = Polygon(np.array([-3.5,2,-3]),np.array([3.5,2,-3]),np.array([3.5,2,3]),np.array([-3.5,2,3]),np.array([-3.5,2,-3]), color = YELLOW_C, fill_color = YELLOW_A, fill_opacity = 0.2)
+
+ plane_y = Polygon(np.array([2,-3.5,-3]),np.array([2,3.5,-3]),np.array([2,3.5,3]),np.array([2,-3.5,3]),np.array([2,-3.5,-3]), color = GREEN_C, fill_color = GREEN_A, fill_opacity = 0.2)
+
+ text_x = TextMobject(r"$x$", "is fixed on this" ,"plane").scale(0.7).to_corner(UL)
+ text_y = TextMobject(r"$y$", "is fixed on this" ,"plane").scale(0.7).to_corner(UR)
+
+ text_x[0].set_color(RED_C)
+ text_y[0].set_color(PINK)
+ text_x[1].set_color(BLUE_C)
+ text_y[1].set_color(BLUE_C)
+ text_x[2].set_color(GREEN_C)
+ text_y[2].set_color(YELLOW_C)
+
+ self.add_fixed_in_frame_mobjects(text_x, text_y)
+
+ for i in range(2,-4,-1):
+
+ parabola_x =ParametricFunction(lambda u : np.array([u,i,(u*u + 3*i*i)*np.exp(1- u*u - i*i)]),color=RED_C,t_min=-3.5,t_max=3.5,)
+
+ parabola_y =ParametricFunction(lambda u : np.array([i,u,(i*i + 3*u*u)*np.exp(1- u*u - i*i)]),color=PINK,t_min=-3.5,t_max=3.5,)
+
+ if(i==2):
+ self.play(ShowCreation(plane_x), ShowCreation(plane_y))
+ parabola_copy_x = parabola_x.copy()
+ parabola_copy_y = parabola_y.copy()
+
+
+ self.play(ShowCreation(parabola_copy_x), ShowCreation(parabola_copy_y))
+ self.wait()
+ self.play(FadeOut(parabola_copy_x), FadeOut(parabola_copy_y))
+
+ else:
+ self.play(ApplyMethod(plane_x.move_to, np.array([0,i,0])),ReplacementTransform(parabola_copy_x, parabola_x),ApplyMethod(plane_y.move_to, np.array([i,0,0])),ReplacementTransform(parabola_copy_y, parabola_y))
+ self.play(FadeOut(parabola_x), FadeOut(parabola_y))
+ self.wait()
+
+ parabola_copy_x = parabola_x.copy()
+ parabola_copy_y = parabola_y.copy()
+
+ self.play(FadeOut(plane_x), FadeOut(plane_y), FadeOut(text_x), FadeOut(text_y))
+
+
+ self.add_fixed_in_frame_mobjects(surface_eqn)
+
+ self.move_camera(phi=80 * DEGREES, theta = 95*DEGREES)
+
+ self.play(ShowCreation(plane1))
+ self.add_fixed_in_frame_mobjects(plane_text_x)
+ self.wait()
+ self.play(ReplacementTransform(paraboloid, paraboloid_x))
+ self.play(FadeOut(plane1), FadeOut(plane_text_x))
+
+ line_x = Line(np.array([-3.5,0,0]), np.array([3.5,0,0]), color = YELLOW_E)
+
+ self.play(ShowCreation(parabola_x_out), ShowCreation(line_x))
+
+ slope_text_x = TexMobject("Slope =", "{\\partial", "f", "\\over", "\\partial", "x}").scale(0.6).move_to(2*UP + 3.5*RIGHT)
+ slope_text_x[0].set_color(ORANGE)
+ slope_text_x.set_color_by_tex("\\partial",GREEN_E)
+ slope_text_x.set_color_by_tex("f",RED_E)
+ slope_text_x[5].set_color(YELLOW_E)
+
+ self.add_fixed_in_frame_mobjects(slope_text_x)
+
+
+ dot_x = Dot().rotate(PI/2).set_color(YELLOW_E)
+ alpha_x = ValueTracker(0)
+ vector_x = self.get_tangent_vector(alpha_x.get_value(),parabola_x_out,scale=1.5)
+ dot_x.add_updater(lambda m: m.move_to(vector_x.get_center()))
+ self.play(
+ ShowCreation(parabola_x_out),
+ GrowFromCenter(dot_x),
+ GrowArrow(vector_x)
+ )
+ vector_x.add_updater(
+ lambda m: m.become(
+ self.get_tangent_vector(alpha_x.get_value()%1,parabola_x_out,scale=1.5)
+ )
+ )
+ self.add(vector_x,dot_x)
+ self.play(alpha_x.increment_value, 1, run_time=10, rate_func=linear)
+
+ self.wait(2)
+ self.play(FadeOut(parabola_x_out), FadeOut(line_x), FadeOut(vector_x), FadeOut(dot_x), FadeOut(slope_text_x))
+
+ self.move_camera(phi=80* DEGREES,theta= 5*DEGREES)
+ self.play(ReplacementTransform(paraboloid_x, paraboloid_copy1))
+ self.wait()
+
+
+
+ self.play(ShowCreation(plane2))
+ self.add_fixed_in_frame_mobjects(plane_text_y)
+ self.wait()
+ self.play(ReplacementTransform(paraboloid_copy1, paraboloid_y))
+ self.play(FadeOut(plane2), FadeOut(plane_text_y))
+
+ line_y = Line(np.array([0,-3.5,0]), np.array([0,3.5,0]), color = GREEN_E)
+
+ self.play(ShowCreation(parabola_y_out), ShowCreation(line_y))
+
+ slope_text_y = TexMobject("Slope =", "{\\partial", "f", "\\over", "\\partial", "y}").scale(0.6).move_to(2*UP + 3.5*RIGHT)
+ slope_text_y[0].set_color(ORANGE)
+ slope_text_y.set_color_by_tex("\\partial",YELLOW_E)
+ slope_text_y.set_color_by_tex("f",PINK)
+ slope_text_y[5].set_color(GREEN_E)
+
+ self.add_fixed_in_frame_mobjects(slope_text_y)
+
+
+ dot_y = Dot().rotate(PI/2).set_color(GREEN_E)
+ alpha_y = ValueTracker(0)
+ vector_y = self.get_tangent_vector(alpha_y.get_value(),parabola_y_out,scale=1.5)
+ dot_y.add_updater(lambda m: m.move_to(vector_y.get_center()))
+ self.play(
+ ShowCreation(parabola_y_out),
+ GrowFromCenter(dot_y),
+ GrowArrow(vector_y)
+ )
+ vector_y.add_updater(
+ lambda m: m.become(
+ self.get_tangent_vector(alpha_y.get_value()%1,parabola_y_out,scale=1.5)
+ )
+ )
+ self.add(vector_y,dot_y)
+ self.play(alpha_y.increment_value, 1, run_time=10, rate_func=linear)
+
+ self.wait(2)
+ self.play(FadeOut(parabola_y_out), FadeOut(line_y), FadeOut(vector_y), FadeOut(dot_y), FadeOut(slope_text_y))
+
+ self.move_camera(phi=60* DEGREES,theta= 45*DEGREES)
+ self.play(ReplacementTransform(paraboloid_y, paraboloid_copy2))
+ self.wait()
+
+
+
+
+
+
+
+
+
+ def get_tangent_vector(self, proportion, curve, dx=0.001, scale=1):
+ coord_i = curve.point_from_proportion(proportion)
+ coord_f = curve.point_from_proportion(proportion + dx)
+ reference_line = Line(coord_i,coord_f)
+ unit_vector = reference_line.get_unit_vector() * scale
+ vector = Line(coord_i - unit_vector , coord_i + unit_vector, color = ORANGE, buff=0)
+ return vector
+
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file6_clariant_rule.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file6_clariant_rule.py new file mode 100644 index 0000000..b79f77c --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file6_clariant_rule.py @@ -0,0 +1,64 @@ +from manimlib.imports import *
+
+class ClariantRule(Scene):
+ def construct(self):
+ derivatives = TextMobject(r"$cos(x)y^3$",r"$-sin(x)y^3$", r"$3cos(x)y^2$", r"$-cos(x)y^3$", r"$-3sin(x)y^2$", r"$-3sin(x)y^2$", r"$6cos(x)y$")
+
+ partial_derivatives = TextMobject(r"$\frac{\partial}{\partial x}$", r"$\frac{\partial}{\partial y}$")
+
+
+ derivatives[0].move_to(2*UP).set_color(PURPLE)
+ derivatives[1].move_to(3*LEFT).set_color(YELLOW_C)
+ derivatives[2].move_to(3*RIGHT).set_color(BLUE_C)
+
+ arrrow_1 = Arrow(derivatives[0].get_bottom(), derivatives[1].get_top())
+ arrrow_1_lab = partial_derivatives[0].copy().scale(0.7)
+ arrrow_1_lab.move_to(2.5*LEFT+ 1.3*UP)
+
+ arrrow_2 = Arrow(derivatives[0].get_bottom(), derivatives[2].get_top())
+ arrrow_2_lab = partial_derivatives[1].copy().scale(0.7)
+ arrrow_2_lab.move_to(2.5*RIGHT+ 1.3*UP)
+
+ self.play(Write(derivatives[0]))
+ self.play(GrowArrow(arrrow_1), GrowArrow(arrrow_2), Write(arrrow_1_lab), Write(arrrow_2_lab))
+
+ self.play(Write(derivatives[1]))
+ self.play(Write(derivatives[2]))
+
+ derivatives[3].move_to(2*DOWN + 4.5*LEFT).set_color(GREEN_C)
+ derivatives[4].move_to(2*DOWN + 1.5*LEFT).set_color(PINK)
+ derivatives[5].move_to(2*DOWN + 1.5*RIGHT).set_color(PINK)
+ derivatives[6].move_to(2*DOWN + 4.5*RIGHT).set_color(ORANGE)
+
+ arrrow_3 = Arrow(derivatives[1].get_bottom(), derivatives[3].get_top())
+ arrrow_3_lab = partial_derivatives[0].copy().scale(0.7)
+ arrrow_3_lab.move_to(4.3*LEFT+ 0.8*DOWN)
+
+ arrrow_4 = Arrow(derivatives[1].get_bottom(), derivatives[4].get_top())
+ arrrow_4_lab = partial_derivatives[1].copy().scale(0.7)
+ arrrow_4_lab.move_to(1.6*LEFT+ 0.8*DOWN)
+
+ arrrow_5 = Arrow(derivatives[2].get_bottom(), derivatives[5].get_top())
+ arrrow_5_lab = partial_derivatives[0].copy().scale(0.7)
+ arrrow_5_lab.move_to(1.6*RIGHT+ 0.8*DOWN)
+
+ arrrow_6 = Arrow(derivatives[2].get_bottom(), derivatives[6].get_top())
+ arrrow_6_lab = partial_derivatives[1].copy().scale(0.7)
+ arrrow_6_lab.move_to(4.3*RIGHT+ 0.8*DOWN)
+
+ self.play(GrowArrow(arrrow_3), GrowArrow(arrrow_4), Write(arrrow_3_lab), Write(arrrow_4_lab))
+ self.play(Write(derivatives[3]), Write(derivatives[4]))
+
+ self.play(GrowArrow(arrrow_5), GrowArrow(arrrow_6), Write(arrrow_5_lab), Write(arrrow_6_lab))
+ self.play(Write(derivatives[5]), Write(derivatives[6]))
+
+ brace1 = Brace(derivatives[4:6], DOWN, buff = SMALL_BUFF, color = RED_C)
+ brace_t1 = brace1.get_text("Mixed partial derivatives are the same!")
+ brace_t1.set_color(RED_C)
+
+ self.play(GrowFromCenter(brace1), FadeIn(brace_t1))
+
+ self.wait()
+
+
+
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file7_partial_deriv_clariant_rule.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file7_partial_deriv_clariant_rule.py new file mode 100644 index 0000000..313c6cd --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file7_partial_deriv_clariant_rule.py @@ -0,0 +1,108 @@ +from manimlib.imports import *
+
+class ClariantRule(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ function = ParametricSurface(
+ lambda u, v: np.array([
+ 3.5*np.sin(u)*np.cos(v),
+ 3.5*np.sin(u)*np.sin(v),
+ 3.5*3.5*np.sin(u)*np.sin(u)*(1+2*np.sin(v)*np.sin(v))*np.exp(1 - 3.5*3.5*np.sin(u)*np.sin(u) )
+ ]),u_min=0,u_max=PI,v_min=0,v_max=2*PI, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1,
+ resolution=(15, 32)).scale(1)
+
+
+ function_copy1 = function.copy()
+ function_copy2 = function.copy()
+
+ func_x =ParametricFunction(
+ lambda u : np.array([
+ u,
+ -1,
+ (u*u )*np.exp(1-u*u)
+ ]),color=RED_E,t_min=-3.5,t_max=3.5,
+ )
+
+ func_y =ParametricFunction(
+ lambda u : np.array([
+ 0,
+ u,
+ (3*u*u)*np.exp(1-u*u)
+ ]),color=PINK,t_min=-3.5,t_max=3.5,
+ )
+
+ plane_x = Polygon(np.array([-3.5,-1,-3]),np.array([3.5,-1,-3]),np.array([3.5,-1,3]),np.array([-3.5,-1,3]),np.array([-3.5,-1,-3]), color = YELLOW_E, fill_color = YELLOW_B, fill_opacity = 0.1)
+ plane_text_x = TextMobject(r"$y = -1$", color = YELLOW_C).move_to(np.array([5,0,2.7])).scale(0.7)
+
+ plane_y = Polygon(np.array([0,-3.5,-3]),np.array([0,3.5,-3]),np.array([0,3.5,3]),np.array([0,-3.5,3]),np.array([0,-3.5,-3]), color = GREEN_E, fill_color = GREEN_B, fill_opacity = 0.1)
+ plane_text_y = TextMobject(r"$x = 0$", color = GREEN_C).move_to(np.array([0,4,2.7])).scale(0.7)
+
+ surface_eqn = TextMobject("Surface", r"$z = (x^2 + 3y^2)e^{(1 - x^2 - y^2)}$", color = YELLOW_C).scale(0.6).move_to(np.array([4.6*LEFT+3.5*UP]))
+ surface_eqn[0].set_color(BLUE_C)
+
+ self.set_camera_orientation(phi=60 * DEGREES, theta = 45*DEGREES)
+
+ self.add(axes)
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.play(ShowCreation(function))
+
+ self.add_fixed_in_frame_mobjects(surface_eqn)
+
+ self.play(ShowCreation(plane_x), ShowCreation(plane_y))
+ self.add_fixed_orientation_mobjects(plane_text_x, plane_text_y)
+
+ self.play(ShowCreation(func_x), ShowCreation(func_y))
+
+ dot_x = Dot().rotate(PI/2).set_color(YELLOW_E)
+ alpha_x = ValueTracker(0)
+ vector_x = self.get_tangent_vector(alpha_x.get_value(),func_x,scale=1.5)
+ dot_x.add_updater(lambda m: m.move_to(vector_x.get_center()))
+ self.play(
+ ShowCreation(func_x),
+ GrowFromCenter(dot_x),
+ GrowArrow(vector_x)
+ )
+ vector_x.add_updater(
+ lambda m: m.become(
+ self.get_tangent_vector(alpha_x.get_value()%1,func_x,scale=1.5)
+ )
+ )
+ dot_y = Dot().rotate(PI/2).set_color(GREEN_E)
+ alpha_y = ValueTracker(0)
+ vector_y = self.get_tangent_vector(alpha_y.get_value(),func_y,scale=1.5)
+ dot_y.add_updater(lambda m: m.move_to(vector_y.get_center()))
+ self.play(
+ ShowCreation(func_y),
+ GrowFromCenter(dot_y),
+ GrowArrow(vector_y)
+ )
+ vector_y.add_updater(
+ lambda m: m.become(
+ self.get_tangent_vector(alpha_y.get_value()%1,func_y,scale=1.5)
+ )
+ )
+ self.add(vector_x,dot_x)
+
+ self.play(alpha_x.increment_value, 1, run_time=10, rate_func=linear)
+
+ self.add(vector_y,dot_y)
+ self.play(alpha_y.increment_value, 1, run_time=10, rate_func=linear)
+
+ self.wait(2)
+
+
+
+
+
+
+
+
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file8_chain_rule.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file8_chain_rule.py new file mode 100644 index 0000000..f50d2d1 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/file8_chain_rule.py @@ -0,0 +1,60 @@ +from manimlib.imports import *
+
+class ChainRule(Scene):
+ def construct(self):
+
+ chain_rule = TextMobject(r"$\frac{dw}{dt}$", r"=", r"$\frac{\partial w}{\partial x}$", r"$\frac{dx}{dt}$", r"+", r"$\frac{\partial w}{\partial y}$", r"$\frac{dy}{dt}$").move_to(4*RIGHT).scale(0.8)
+
+ chain_rule[0].set_color(ORANGE)
+ chain_rule[2].set_color(GREEN_C)
+ chain_rule[3].set_color(RED_C)
+ chain_rule[5].set_color(YELLOW_C)
+ chain_rule[6].set_color(BLUE_C)
+
+ functions = TextMobject(r"$w =f(x,y)$",r"$x$", r"$y$", r"$t$")
+
+ functions[0].move_to(3.3*UP+1*LEFT).set_color(ORANGE)
+ functions[1].move_to(3.3*LEFT).set_color(PURPLE)
+ functions[2].move_to(1.3*RIGHT).set_color(PURPLE)
+ functions[3].move_to(3.3*DOWN+1*LEFT).set_color(WHITE)
+
+ partial_derivatives = TextMobject(r"$\frac{\partial w}{\partial x}$", r"$\frac{\partial w}{\partial y}$")
+
+ partial_derivatives[0].move_to(1.5*UP+3*LEFT).set_color(GREEN_C)
+ partial_derivatives[1].move_to(1.5*UP+1*RIGHT).set_color(YELLOW_C)
+
+ derivatives = TextMobject(r"$\frac{dx}{dt}$", r"$\frac{dy}{dt}$")
+
+ derivatives[0].move_to(1.5*DOWN+3*LEFT).set_color(RED_C)
+ derivatives[1].move_to(1.5*DOWN+1*RIGHT).set_color(BLUE_C)
+
+ line_f_x = Line(np.array([-1,3,0]), np.array([-3,0,0]), color = BLUE_C)
+ line_f_y = Line(np.array([-1,3,0]), np.array([1,0,0]), color = BLUE_C)
+ line_x_t = Line(np.array([-3,0,0]), np.array([-1,-3,0]), color = BLUE_C)
+ line_y_t = Line(np.array([1,0,0]), np.array([-1,-3,0]), color = BLUE_C)
+
+ dot_f = Dot().shift(np.array([-1,3,0])).set_color(BLUE_C)
+ dot_x = Dot().shift(np.array([-3,0,0])).set_color(BLUE_C)
+ dot_y = Dot().shift(np.array([1,0,0])).set_color(BLUE_C)
+ dot_t = Dot().shift(np.array([-1,-3,0])).set_color(BLUE_C)
+
+ variables = TextMobject("Dependent Variable","Intermediate Variables", "Dependent Variable").set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE).scale(0.7)
+ variables[0].move_to(3.3*UP+3.5*RIGHT)
+ variables[1].move_to(3.5*RIGHT)
+ variables[2].move_to(3.3*DOWN+3.5*RIGHT)
+
+ self.play(ShowCreation(dot_f), Write(functions[0]))
+ self.play(ShowCreation(dot_x), ShowCreation(line_f_x), Write(functions[1]), ShowCreation(dot_y), ShowCreation(line_f_y), Write(functions[2]))
+ self.play(Write(partial_derivatives[0]), Write(partial_derivatives[1]))
+ self.wait()
+
+ self.play(ShowCreation(dot_t), ShowCreation(line_x_t), ShowCreation(line_y_t), Write(functions[3]))
+ self.play(Write(derivatives[0]), Write(derivatives[1]))
+ self.wait()
+
+ self.play(Write(variables[0]), Write(variables[1]), Write(variables[2]))
+
+ self.play(FadeOut(variables))
+ self.play(Write(chain_rule))
+ self.wait()
+
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file1_partial_deriv_gas_law.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file1_partial_deriv_gas_law.gif Binary files differnew file mode 100644 index 0000000..560a7c0 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file1_partial_deriv_gas_law.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file2_partial_deriv_hill.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file2_partial_deriv_hill.gif Binary files differnew file mode 100644 index 0000000..f4c3f49 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file2_partial_deriv_hill.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file3_partial_deriv_defn.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file3_partial_deriv_defn.gif Binary files differnew file mode 100644 index 0000000..e0e42db --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file3_partial_deriv_defn.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file4_partial_deriv_example.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file4_partial_deriv_example.gif Binary files differnew file mode 100644 index 0000000..30682cb --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file4_partial_deriv_example.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file5_partial_deriv_func_2maximas.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file5_partial_deriv_func_2maximas.gif Binary files differnew file mode 100644 index 0000000..aa74437 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file5_partial_deriv_func_2maximas.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file6_clariant_rule.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file6_clariant_rule.gif Binary files differnew file mode 100644 index 0000000..8377827 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file6_clariant_rule.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file7_partial_deriv_clariant_rule.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file7_partial_deriv_clariant_rule.gif Binary files differnew file mode 100644 index 0000000..ecef499 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file7_partial_deriv_clariant_rule.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file8_chain_rule.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file8_chain_rule.gif Binary files differnew file mode 100644 index 0000000..596b08d --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file8_chain_rule.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/README.md b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/README.md new file mode 100644 index 0000000..4339c30 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/README.md @@ -0,0 +1,20 @@ +**file1_scalar_function** + + +**file2_domain_range** + + +**file3_parabola_example** + + +**file4_level_curves** + + +**file5_level_surface** + + +**file6_scalar_function_application** + + +**file7_neural_nets** + diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/Scalar_Function_Quiz.pdf b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/Scalar_Function_Quiz.pdf Binary files differindex 6d94a2c..6d94a2c 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/Scalar_Function_Quiz.pdf +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/Scalar_Function_Quiz.pdf diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file1_scalar_functions.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file1_scalar_functions.py new file mode 100644 index 0000000..1a6f4ed --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file1_scalar_functions.py @@ -0,0 +1,50 @@ +from manimlib.imports import *
+
+class ScalarFunction(Scene):
+ def construct(self):
+ circle = Circle(radius = 1.5, color = BLUE_E, fill_color = BLUE_C, fill_opacity = 0.1).move_to(2*LEFT)
+ dot_circle = Dot().shift(np.array([-1.5,0,0])).set_color(BLUE_E)
+ dot_circle_lab = TextMobject(r"$a$", color = BLUE_E).next_to(dot_circle, DOWN)
+
+ arrow = Arrow(np.array([3,-3,0]),np.array([3,3,0]))
+ line = Line(np.array([3,-1.5,0]),np.array([3,1.5,0]), color = RED_C)
+
+ dot0 = Dot().shift(np.array([3,0,0])).set_color(RED_E)
+ dot0_lab = TextMobject(r"$f(a)$", color = RED_E).scale(0.8).next_to(dot0, RIGHT)
+
+ dot1 = Dot().shift(np.array([3,-1.5,0])).set_color(RED_C)
+
+ dot2 = Dot().shift(np.array([3,1.5,0])).set_color(RED_C)
+ dot2_lab = TextMobject(r"$f(A)$", color = RED_C).scale(0.8).next_to(dot2, RIGHT)
+
+ arrow_f = Arrow(np.array([-1.5,0,0]),np.array([3,0,0]), color = YELLOW_C, buff = 0.1)
+
+ R = TextMobject(r"$\mathbb{R}$", color = WHITE).move_to(np.array([3,-3.3,0]))
+
+ A = TextMobject(r"$A$", color = BLUE_E).move_to(np.array([-2.5,-3.3,0]))
+
+ F = TextMobject(r"$f$", color = GREY).move_to(np.array([0,-2.9,0]))
+
+ F_center = TextMobject(r"$f$", color = YELLOW_C).move_to(np.array([0.8,0.5,0]))
+
+ arrow_R_A = Arrow(np.array([-2.3,-3.3,0]),np.array([2.7,-3.3,0]), color = GREY, buff = 0.1)
+
+ scalar_function = TextMobject(r"Scalar Valued Function", r"$f: A \rightarrow \mathbb{R}$", color = PURPLE).move_to(np.array([0,3.5,0]))
+ scalar_function[1].set_color(GREEN_C)
+
+
+
+ self.play(ShowCreation(circle))
+ self.play(ShowCreation(arrow))
+
+
+ self.play(ShowCreation(dot1), ShowCreation(dot2))
+ self.play(ShowCreation(dot_circle))
+ self.play(ShowCreation(dot_circle_lab), ShowCreation(dot2_lab))
+ self.play(ShowCreation(A), ShowCreation(R))
+ self.play(GrowArrow(arrow_f), ShowCreation(dot0), ShowCreation(dot0_lab), ShowCreation(F_center), GrowArrow(arrow_R_A), ShowCreation(F), Transform(circle.copy(), line.copy()))
+
+ self.play(Write(scalar_function))
+
+
+ self.wait(2)
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/scalar_function_domain_range.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file2_domain_range.py index 9b1ca7b..1b54cb6 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/scalar_function_domain_range.py +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file2_domain_range.py @@ -20,6 +20,11 @@ class PlotGraphs(GraphScene): self.play(FadeOut(topic))
self.wait(1)
+ scalar_func_R = TextMobject(r"Scalar Valued Functions in $R$").scale(1.5).set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+ self.play(Write(scalar_func_R))
+ self.play(FadeOut(scalar_func_R))
+ self.wait(1)
+
XTD = self.x_axis_width/(self.x_max- self.x_min)
YTD = self.y_axis_height/(self.y_max- self.y_min)
@@ -40,8 +45,8 @@ class PlotGraphs(GraphScene): rangeMsg.scale(0.5)
rangeMsg.set_color(YELLOW)
- domainline1 = Line(self.graph_origin+0.6*YTD*DOWN+1.2*XTD*LEFT, self.graph_origin+0.6*YTD*DOWN + 4*XTD*LEFT)
- domainline2 = Arrow(self.graph_origin+0.6*YTD*DOWN+1.1*XTD*RIGHT, self.graph_origin+0.6*YTD*DOWN + 5.3*XTD*RIGHT)
+ domainline1 = Arrow(self.graph_origin+0.6*YTD*DOWN+1.2*XTD*LEFT, self.graph_origin+0.6*YTD*DOWN + 4*XTD*LEFT, buff = 0.1)
+ domainline2 = Arrow(self.graph_origin+0.6*YTD*DOWN+1.1*XTD*RIGHT, self.graph_origin+0.6*YTD*DOWN + 5.3*XTD*RIGHT, buff = 0.1)
domainline1.set_color(PINK)
domainline2.set_color(PINK)
@@ -128,5 +133,58 @@ class PlotSineGraphs(GraphScene): self.play(Write(domainMsg))
self.wait(3)
+
+
+
+class Paraboloid(ThreeDScene):
+ def construct(self):
+
+ scalar_func_R2 = TextMobject(r"Scalar Valued Functions in $R^2$").scale(1.5).set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+ self.play(Write(scalar_func_R2))
+ self.play(FadeOut(scalar_func_R2))
+ self.wait(1)
+
+ axes = ThreeDAxes()
+
+ paraboloid = ParametricSurface(
+ lambda u, v: np.array([
+ 2*np.sin(u)*np.cos(v),
+ 2*np.sin(u)*np.sin(v),
+ 2*2*np.sin(u)*np.sin(u)
+ ]),u_min=0,u_max=PI/2,v_min=0,v_max=2*PI,checkerboard_colors=[GREEN_C, GREEN_E],
+ resolution=(15, 32)).scale(1)
+
+ domain = Polygon(np.array([-5,-5,0]),np.array([5,-5,0]),np.array([5,5,0]),np.array([-5,5,0]),np.array([-5,-5,0]), color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.2)
+ domain_lab = TextMobject(r"$Domain: R^2$", color = YELLOW_C).scale(0.7).move_to(1*DOWN + 2*LEFT)
+
+ rangef = Line(np.array([0, 0,0]), np.array([0, 0,5]), color = RED_C)
+ rangef_lab = TextMobject(r"$Range: z \geq 0$", color = RED_C).scale(0.7).move_to(2*UP + 1.5*RIGHT)
+
+ func = TextMobject(r"$z = f(x,y) = x^2+y^2$").scale(0.7).move_to(3*UP + 4*LEFT).set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+
+ self.set_camera_orientation(phi=60 * DEGREES, theta = 0*DEGREES)
+ self.begin_ambient_camera_rotation(rate=0.3)
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+
+
+ self.add_fixed_in_frame_mobjects(func)
+ self.play(Write(paraboloid))
+ self.play(ShowCreation(domain))
+ self.add_fixed_in_frame_mobjects(domain_lab)
+ self.wait()
+ self.play(ShowCreation(rangef))
+ self.add_fixed_in_frame_mobjects(rangef_lab)
+ self.wait(5)
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/scalar_function_parabola_example.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file3_parabola_example.py index 74dc063..63c16b3 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/scalar_function_parabola_example.py +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file3_parabola_example.py @@ -9,11 +9,10 @@ class Parabola(ThreeDScene): 2*np.cosh(u)*np.cos(v),
2*np.cosh(u)*np.sin(v),
2*np.sinh(u)
- ]),v_min=0,v_max=TAU,u_min=0,u_max=2,checkerboard_colors=[YELLOW_D, YELLOW_E],
+ ]),v_min=0,v_max=TAU,u_min=0,u_max=2,checkerboard_colors=[YELLOW_D, YELLOW_E],#
resolution=(15, 32))
- text3d = TextMobject(r"Plot of $f: \mathbb{R}^2 \rightarrow \mathbb{R}$", "z = f(x,y)")
- self.add_fixed_in_frame_mobjects(text3d)
+ text3d = TextMobject(r"Plot of $f: \mathbb{R}^2 \rightarrow \mathbb{R}$", r"$z = f(x,y) = \sqrt{x^2 + y^2 - 4}$")
text3d[0].move_to(4*LEFT+2*DOWN)
text3d[1].next_to(text3d[0], DOWN)
text3d[0].set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
@@ -21,8 +20,21 @@ class Parabola(ThreeDScene): #self.set_camera_orientation(phi=0 * DEGREES,theta=270*DEGREES)
self.move_camera(phi=110* DEGREES,theta=45*DEGREES)
+
self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+
self.play(ShowCreation(paraboloid))
+ self.add_fixed_in_frame_mobjects(text3d)
self.play(Write(text3d[0]))
self.play(Write(text3d[1]))
self.begin_ambient_camera_rotation(rate=0.2)
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file4_level_curves.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file4_level_curves.py new file mode 100644 index 0000000..2b6f719 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file4_level_curves.py @@ -0,0 +1,118 @@ +from manimlib.imports import *
+
+class LevelCurves(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ paraboloid = ParametricSurface(
+ lambda u, v: np.array([
+ u*np.cos(v),
+ u*np.sin(v),
+ -u*u+2
+ ]),u_min=-1.414,u_max=1.414,v_min=0,v_max=2*PI, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1,
+ resolution=(15, 32)).scale(1)
+
+ plane_0 = Polygon(np.array([2,-2,0]),np.array([2,2,0]),np.array([-2,2,0]),np.array([-2,-2,0]),np.array([2,-2,0]), color = BLUE_E, fill_color = BLUE_E, fill_opacity = 0.3)
+ plane_0_lab = TextMobject("C = 0").move_to(0.4*UP+3.2*RIGHT).set_color(BLUE_E).scale(0.6)
+ circle_0 = Circle(radius = 1.414 , color = BLUE_E)
+ circle_0_lab = TextMobject("0").move_to(1.1*DOWN+1.1*RIGHT).set_color(BLUE_E).scale(0.6)
+
+ plane_0_5 = Polygon(np.array([2,-2,0.5]),np.array([2,2,0.5]),np.array([-2,2,0.5]),np.array([-2,-2,0.5]),np.array([2,-2,0.5]), color = GREEN_C, fill_color = GREEN_C, fill_opacity = 0.3)
+ plane_0_5_lab = TextMobject("C = 0.5").move_to(0.8*UP+3.4*RIGHT).set_color(GREEN_C).scale(0.6)
+ circle_0_5 = Circle(radius = 1.224 , color = GREEN_C)
+ circle_0_5_lab = TextMobject("0.5").move_to(0.9*DOWN+0.9*RIGHT).set_color(GREEN_C).scale(0.6)
+ circle_0_5_copy = circle_0_5.copy().move_to(np.array([0,0,0.5]))
+
+ plane_1 = Polygon(np.array([2,-2,1]),np.array([2,2,1]),np.array([-2,2,1]),np.array([-2,-2,1]),np.array([2,-2,1]), color = YELLOW_C, fill_color = YELLOW_C, fill_opacity = 0.3)
+ plane_1_lab = TextMobject("C = 1").move_to(1.2*UP+3.3*RIGHT).set_color(YELLOW_C).scale(0.6)
+ circle_1 = Circle(radius = 1 , color = YELLOW_C)
+ circle_1_lab = TextMobject("1").move_to(0.7*DOWN+0.7*RIGHT).set_color(YELLOW_C).scale(0.6)
+ circle_1_copy = circle_1.copy().move_to(np.array([0,0,1]))
+
+ plane_1_5 = Polygon(np.array([2,-2,1.5]),np.array([2,2,1.5]),np.array([-2,2,1.5]),np.array([-2,-2,1.5]),np.array([2,-2,1.5]), color = ORANGE, fill_color = ORANGE, fill_opacity = 0.3)
+ plane_1_5_lab = TextMobject("C = 1.5").move_to(1.7*UP+3.4*RIGHT).set_color(ORANGE).scale(0.6)
+ circle_1_5 = Circle(radius = 0.707 , color = ORANGE)
+ circle_1_5_lab = TextMobject("1.5").move_to(0.5*DOWN+0.5*RIGHT).set_color(ORANGE).scale(0.6)
+ circle_1_5_copy = circle_1_5.copy().move_to(np.array([0,0,1.5]))
+
+ plane_2 = Polygon(np.array([2,-2,2]),np.array([2,2,2]),np.array([-2,2,2]),np.array([-2,-2,2]),np.array([2,-2,2]), color = RED_C, fill_color = RED_C, fill_opacity = 0.3)
+ plane_2_lab = TextMobject("C = 2").move_to(2.1*UP+3.3*RIGHT).set_color(RED_C).scale(0.6)
+ dot_2 = Dot().set_fill(RED_C)
+ circle_2_lab = TextMobject("2").move_to(0.2*DOWN+0.2*RIGHT).set_color(RED_C).scale(0.6)
+ dot_2_copy = dot_2.copy().move_to(np.array([0,0,2]))
+
+ level_curves_line1 = DashedLine(np.array([0,-1.414,0]),np.array([0,-2,1]), color = WHITE)
+ level_curves_line2 = DashedLine(np.array([0,-1.224,0.5]),np.array([0,-2,1]), color = WHITE)
+ level_curves_line3 = DashedLine(np.array([0,-1,1]),np.array([0,-2,1]), color = WHITE)
+ level_curves_line4 = DashedLine(np.array([0,-0.707,1.5]),np.array([0,-2,1]), color = WHITE)
+ level_curves_line5 = DashedLine(np.array([0,0,2]),np.array([0,-2,1]), color = WHITE)
+
+ level_curves = TextMobject("Level Curves").move_to(1.4*UP+3*LEFT).set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE).scale(0.8)
+
+
+ contour_line1 = DashedLine(np.array([0,-1.414,0]),np.array([0,-2,1]), color = WHITE)
+ contour_line2 = DashedLine(np.array([0,-1.224,0]),np.array([0,-2,1]), color = WHITE)
+ contour_line3 = DashedLine(np.array([0,-1,0]),np.array([0,-2,1]), color = WHITE)
+ contour_line4 = DashedLine(np.array([0,-0.707,0]),np.array([0,-2,1]), color = WHITE)
+ contour_line5 = DashedLine(np.array([0,0,0]),np.array([0,-2,1]), color = WHITE)
+
+ contours = TextMobject("Contours").move_to(1.4*UP+2.7*LEFT).set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE).scale(0.8)
+
+
+ topic = TextMobject("Contour Plot").move_to(3*UP+3*LEFT).set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE).scale(0.8)
+
+ self.set_camera_orientation(phi=80 * DEGREES, theta = 0*DEGREES)
+ #self.set_camera_orientation(phi=0 * DEGREES, theta = 0*DEGREES)
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.play(Write(paraboloid))
+ self.wait()
+ self.play(ShowCreation(plane_0), ShowCreation(circle_0))
+ self.add_fixed_in_frame_mobjects(plane_0_lab)
+ self.wait()
+ self.play(ShowCreation(plane_0_5), ShowCreation(circle_0_5_copy), ShowCreation(circle_0_5))
+ self.add_fixed_in_frame_mobjects(plane_0_5_lab)
+ self.wait()
+ self.play(ShowCreation(plane_1), ShowCreation(circle_1_copy), ShowCreation(circle_1))
+ self.add_fixed_in_frame_mobjects(plane_1_lab)
+ self.wait()
+ self.play(ShowCreation(plane_1_5), ShowCreation(circle_1_5_copy), ShowCreation(circle_1_5))
+ self.add_fixed_in_frame_mobjects(plane_1_5_lab)
+ self.wait()
+ self.play(ShowCreation(plane_2), ShowCreation(dot_2_copy), ShowCreation(dot_2))
+ self.add_fixed_in_frame_mobjects(plane_2_lab)
+ self.wait()
+
+ self.move_camera(phi=60 * DEGREES, theta = 30*DEGREES,run_time=3)
+ self.play(FadeOut(plane_0), FadeOut(plane_0_lab), FadeOut(plane_0_5), FadeOut(plane_0_5_lab), FadeOut(plane_1), FadeOut(plane_1_lab), FadeOut(plane_1_5), FadeOut(plane_1_5_lab), FadeOut(plane_2), FadeOut(plane_2_lab))
+
+ self.play(GrowArrow(level_curves_line1), GrowArrow(level_curves_line2), GrowArrow(level_curves_line3), GrowArrow(level_curves_line4), GrowArrow(level_curves_line5))
+ self.add_fixed_in_frame_mobjects(level_curves)
+ self.wait()
+ self.play(FadeOut(level_curves_line1), FadeOut(level_curves_line2), FadeOut(level_curves_line3), FadeOut(level_curves_line4), FadeOut(level_curves_line5), FadeOut(level_curves))
+ self.play(FadeOut(circle_0_5_copy), FadeOut(circle_1_copy), FadeOut(circle_1_5_copy), FadeOut(dot_2_copy))
+ self.wait()
+
+ self.play(GrowArrow(contour_line1), GrowArrow(contour_line2), GrowArrow(contour_line3), GrowArrow(contour_line4), GrowArrow(contour_line5))
+ self.add_fixed_in_frame_mobjects(contours)
+ self.wait()
+ self.play(FadeOut(contour_line1), FadeOut(contour_line2), FadeOut(contour_line3), FadeOut(contour_line4), FadeOut(contour_line5), FadeOut(contours))
+
+
+ self.move_camera(phi=0 * DEGREES, theta = 0*DEGREES,run_time=3)
+ self.play(FadeOut(paraboloid))
+ self.wait()
+
+ self.add_fixed_in_frame_mobjects(circle_0_lab, circle_0_5_lab, circle_1_lab, circle_1_5_lab,circle_2_lab)
+ self.add_fixed_in_frame_mobjects(topic)
+ self.wait(3)
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file5_level_surface.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file5_level_surface.py new file mode 100644 index 0000000..8052676 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file5_level_surface.py @@ -0,0 +1,78 @@ +from manimlib.imports import *
+
+class LevelSurface(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+
+ surface_0 = ParametricSurface(
+ lambda u, v: np.array([
+ u*np.cos(v),
+ u*np.sin(v),
+ (u*u*np.cos(v)*np.cos(v))-(u*np.sin(v)/5)+0
+ ]),u_min=-1,u_max=1,v_min=0,v_max=2*PI,checkerboard_colors=[RED_C, RED_E],
+ resolution=(15, 32)).scale(1)
+
+ k_0 = TextMobject("K = 0", color = RED_C).scale(0.7)
+
+ surface_1 = ParametricSurface(
+ lambda u, v: np.array([
+ u*np.cos(v),
+ u*np.sin(v),
+ (u*u*np.cos(v)*np.cos(v))-(u*np.sin(v)/5)+1
+ ]),u_min=-1,u_max=1,v_min=0,v_max=2*PI,checkerboard_colors=[GREEN_C, GREEN_E],
+ resolution=(15, 32)).scale(1)
+
+ k_1 = TextMobject("K = 1", color = GREEN_C).scale(0.7)
+
+ surface_2 = ParametricSurface(
+ lambda u, v: np.array([
+ u*np.cos(v),
+ u*np.sin(v),
+ (u*u*np.cos(v)*np.cos(v))-(u*np.sin(v)/5)+2
+ ]),u_min=-1,u_max=1,v_min=0,v_max=2*PI,checkerboard_colors=[YELLOW_C, YELLOW_E],
+ resolution=(15, 32)).scale(1)
+
+ k_2 = TextMobject("K = 2", color = YELLOW_C).scale(0.7)
+
+ func = TextMobject(r"$w = g(x,y,z)$", r"$= z - f(x,y)$", r"$z-x^2+y/5 = K$")
+ func.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE)
+
+ self.set_camera_orientation(phi=90 * DEGREES, theta = 90*DEGREES)
+ self.begin_ambient_camera_rotation(rate=0.3)
+
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(3.7*UP)
+
+ self.add_fixed_in_frame_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
+
+ self.play(Write(surface_0))
+ self.add_fixed_in_frame_mobjects(k_0)
+ k_0.move_to(np.array([1.4*RIGHT ]))
+
+ self.play(Write(surface_1))
+ self.add_fixed_in_frame_mobjects(k_1)
+ k_1.move_to(np.array([1.4*RIGHT + 1*UP]))
+
+ self.play(Write(surface_2))
+ self.add_fixed_in_frame_mobjects(k_2)
+ k_2.move_to(np.array([1.4*RIGHT + 2*UP]))
+ self.wait()
+
+ self.add_fixed_in_frame_mobjects(func)
+ func[0].move_to(np.array([4.5*LEFT + 3*UP]))
+ func[1].move_to(np.array([4.5*LEFT + 2.5*UP]))
+ func[2].move_to(np.array([4.5*LEFT + 2*UP]))
+
+ self.wait(3)
+ self.move_camera(phi=60 * DEGREES,run_time=3)
+ self.wait(2)
+
+
+
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/scalar_function_application.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file6_scalar_function_application.py index 56b3e53..3ccfad6 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/scalar_function_application.py +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file6_scalar_function_application.py @@ -3,12 +3,23 @@ from manimlib.imports import * class ScalarApplication(ThreeDScene):
def construct(self):
axes = ThreeDAxes() # creates a 3D Axis
+
+ self.add(axes)
+
+ axis = TextMobject(r"X",r"Y",r"Z")
+ axis[0].move_to(6*RIGHT)
+ axis[1].move_to(6*UP)
+ axis[2].move_to(np.array([0,0,3.7]))
+
+ self.add_fixed_orientation_mobjects(axis[2])
+ self.add_fixed_orientation_mobjects(axis[0])
+ self.add_fixed_orientation_mobjects(axis[1])
cube = Cube()
- cube.set_fill(YELLOW_E, opacity = 0.1)
+ cube.set_fill(YELLOW_C, opacity = 0.2)
cube.scale(2)
self.set_camera_orientation(phi=0 * DEGREES,theta=270*DEGREES)
- self.play(ShowCreation(cube),ShowCreation(axes))
+ self.play(ShowCreation(cube))
dot = Sphere()
dot.scale(0.1)
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/scalar_function_neural_nets.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file7_neural_nets.py index eb6bf45..eb6bf45 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/scalar_function_neural_nets.py +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/file7_neural_nets.py diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file1_scalar_functions.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file1_scalar_functions.gif Binary files differnew file mode 100644 index 0000000..bea9c7b --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file1_scalar_functions.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file2_domain_range.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file2_domain_range.gif Binary files differnew file mode 100644 index 0000000..6801e4f --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file2_domain_range.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file3_parabola_example.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file3_parabola_example.gif Binary files differnew file mode 100644 index 0000000..af9e536 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file3_parabola_example.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file4_level_curves.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file4_level_curves.gif Binary files differnew file mode 100644 index 0000000..b4ac106 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file4_level_curves.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file5_level_surface.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file5_level_surface.gif Binary files differnew file mode 100644 index 0000000..e4dc80d --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file5_level_surface.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file6_scalar_function_application.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file6_scalar_function_application.gif Binary files differnew file mode 100644 index 0000000..8bb176a --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file6_scalar_function_application.gif diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_neural_nets.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file7_neural_nets.gif Binary files differindex 9d24688..9d24688 100644 --- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/Scalar Functions/gifs/scalar_function_neural_nets.gif +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file7_neural_nets.gif diff --git a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/flux/README.md b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/flux/README.md index e69de29..0af7aa1 100644 --- a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/flux/README.md +++ b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/flux/README.md @@ -0,0 +1,8 @@ +**file1_flux_through_sphere.py** + +**file2_mobius_strip.py** + +**file3_normal_vector.py** + +**file4_cube_surface.py** + diff --git a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/flux/file4_cube_surface.py b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/flux/file4_cube_surface.py index 5963996..9301a00 100644 --- a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/flux/file4_cube_surface.py +++ b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/flux/file4_cube_surface.py @@ -43,29 +43,186 @@ class cuber(ThreeDScene): n1.rotate(PI/2,axis=RIGHT) n1.shift(3*RIGHT+1.3*OUT+DOWN) + spaceloc = [[0,0,2],[1,0,2],[-1,0,2],[2,0,2],[-2,0,2],[3,0,2],[-3,0,2], + [0,1,2],[1,1,2],[-1,1,2],[2,1,2],[-2,1,2],[3,1,2],[-3,1,2], + [0,-1,2],[1,-1,2],[-1,-1,2],[2,-1,2],[-2,-1,2],[3,-1,2],[-3,-1,2], + [0,2,2],[1,2,2],[-1,2,2],[2,2,2],[-2,2,2],[3,2,2],[-3,2,2], + [0,-2,2],[1,-2,2],[-1,-2,2],[2,-2,2],[-2,-2,2],[3,-2,2],[-3,-2,2], + [0,3,2],[1,3,2],[-1,3,2],[2,3,2],[-2,3,2],[3,3,2],[-3,3,2], + [0,3,2],[1,3,2],[-1,3,2],[2,3,2],[-2,3,2],[3,3,2],[-3,3,2], + [0,4,2],[1,4,2],[-1,4,2],[2,4,2],[-2,4,2],[3,4,2],[-3,4,2], + [0,4,2],[1,4,2],[-1,4,2],[2,4,2],[-2,4,2],[3,4,2],[-3,4,2], + [0,5,2],[1,5,2],[-1,5,2],[2,5,2],[-2,5,2],[3,5,2],[-3,5,2], + [0,5,2],[1,5,2],[-1,5,2],[2,5,2],[-2,5,2],[3,5,2],[-3,5,2], + [0,6,2],[1,6,2],[-1,6,2],[2,6,2],[-2,6,2],[3,6,2],[-3,6,2], + [0,1.5,2],[1,1.5,2],[-1,1.5,2],[2,1.5,2],[-2,1.5,2],[3,1.5,2],[-3,1.5,2], + [0,3,2],[1,3,2],[-1,3,2],[2,3,2],[-2,3,2],[3,3,2],[-3,3,2]] + + + veclist1=[Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E)] + + + + + + [veclist1[i].rotate(PI/4,axis=RIGHT) for i in range(10,30,1)] + [veclist1[i].rotate(PI/4,axis=RIGHT) for i in range(40,80,2)] + [veclist1[i].rotate(PI/6,axis=OUT) for i in range(98)] + [veclist1[i].rotate(PI/8,axis=DOWN) for i in range(98)] + vectorfield1=VGroup(*veclist1) + [veclist1[i].shift(spaceloc[i]) for i in range(98)] - self.set_camera_orientation(phi=75 * DEGREES,theta=-15*DEGREES) + + + veclist2=[Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E), + Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E),Vector(color=GOLD_E)] + + + + + + [veclist2[i].rotate(PI/4,axis=RIGHT) for i in range(10,30,1)] + [veclist2[i].rotate(PI/4,axis=RIGHT) for i in range(40,80,2)] + [veclist2[i].rotate(PI/6,axis=OUT) for i in range(98)] + [veclist2[i].rotate(PI/8,axis=DOWN) for i in range(98)] + vectorfield2=VGroup(*veclist2) + [veclist2[i].shift(spaceloc[i]) for i in range(98)] + + + + veclist3=[Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector()] + + + + + + [veclist3[i].rotate(PI/4,axis=RIGHT) for i in range(10,30,1)] + [veclist3[i].rotate(PI/4,axis=RIGHT) for i in range(40,80,2)] + [veclist3[i].rotate(PI/6,axis=OUT) for i in range(98)] + [veclist3[i].rotate(PI/8,axis=DOWN) for i in range(98)] + vectorfield3=VGroup(*veclist3) + [veclist3[i].shift(spaceloc[i]) for i in range(98)] + + + + + veclist4=[Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector(), + Vector(color=RED),Vector(color=GREEN),Vector(color=BLUE),Vector(color=PINK),Vector(color=MAROON),Vector(color=GREEN),Vector()] + + + + + + [veclist4[i].rotate(PI/4,axis=RIGHT) for i in range(10,30,1)] + [veclist4[i].rotate(PI/4,axis=RIGHT) for i in range(40,80,2)] + [veclist4[i].rotate(PI/6,axis=OUT) for i in range(98)] + [veclist4[i].rotate(PI/8,axis=DOWN) for i in range(98)] + vectorfield4=VGroup(*veclist4) + [veclist4[i].shift(spaceloc[i]) for i in range(98)] + + + vectorfield1.shift(1.5*DOWN) + vectorfield2.shift(IN+1.5*DOWN) + vectorfield3.shift(2*IN+1.5*DOWN) + vectorfield4.shift(3*IN+1.5*DOWN) + + vectors=[vectorfield1,vectorfield2,vectorfield3,vectorfield4] + vectorfield=VGroup(*vectors) + vectorfield.scale(1.25) + + fv=[Vector(color=ORANGE),Vector(color=ORANGE),Vector(color=ORANGE),Vector(color=ORANGE), + Vector(color=ORANGE),Vector(color=ORANGE),Vector(color=ORANGE),Vector(color=ORANGE), + Vector(color=ORANGE),Vector(color=ORANGE),Vector(color=ORANGE),Vector(color=ORANGE), + Vector(color=ORANGE),Vector(color=ORANGE),Vector(color=ORANGE),Vector(color=ORANGE), + ] + + spaceloc2 = [[1.5,0.5,0.5],[1.5,1,0.5],[1.5,1.5,0.5],[1.5,2,0.5], + [1.5,0.5,1],[1.5,1,1],[1.5,1.5,1],[1.5,2,1], + [1.5,0.5,1.5],[1.5,1,1.5],[1.5,1.5,1.5],[1.5,2,1.5], + [1.5,0.5,2],[1.5,1,2],[1.5,1.5,2],[1.5,2,2]] + + [fv[i].rotate(PI/4,axis=RIGHT) for i in range(1)] + [fv[i].rotate(PI/6,axis=OUT) for i in range(16)] + [fv[i].rotate(PI/8,axis=DOWN) for i in range(16)] + [fv[i].shift(spaceloc2[i]) for i in range(16)] + fvfield=VGroup(*fv) + fvfield.shift(0.5*IN+2*DOWN) + + flux=TextMobject("Flux through one side of the cube").set_color(ORANGE) + flux.shift(3*UP+1.5*LEFT) + + + + + + self.set_camera_orientation(phi=70 * DEGREES,theta=-75*DEGREES) self.play(ShowCreation(axes),ShowCreation(axis_label)) - self.play(ShowCreation(cube, run_time=3)) - self.begin_ambient_camera_rotation(rate=-0.2) - # self.move_camera(phi=150*DEGREES,theta=-45*DEGREES, run_time=3) + self.play(ShowCreation(vectorfield)) + self.add(fvfield) + self.begin_ambient_camera_rotation(rate=0.01) + + self.play(ShowCreation(cube, run_time=1)) + self.wait(1) self.play(ShowCreation(sq3)) - self.wait(1) - self.play(ShowCreation(v1),ShowCreation(n1)) + self.play(FadeOut(cube)) + self.play(FadeOut(vectorfield)) + self.add_fixed_in_frame_mobjects(flux) + # self.play(ShowCreation(flux)) self.wait(1) - self.stop_ambient_camera_rotation() - self.wait(2) - - - # self.play(Write(t1)) - # self.play(Transform(vg,t1)) - # self.wait(3) - # self.play(ReplacementTransform(t1,t2)) - # self.wait(3) - # # self.move_camera(phi=50*DEGREES,theta=-45*DEGREES,run_time=3) - # self.wait(8) - # self.move_camera(phi=75 * DEGREES, run_time=3) - # self.wait(3) + self.play(ShowCreation(v1),ShowCreation(n1)) + self.wait(6) + # self.stop_ambient_camera_rotation() + diff --git a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/surface-integrals/README.md b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/surface-integrals/README.md index df307cd..a1de8b5 100644 --- a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/surface-integrals/README.md +++ b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/surface-integrals/README.md @@ -1,2 +1,10 @@ -**Test**
-
+**file1_projection.py** + +**file2_cube.py** + +**file3_cube_sideC.py** + +**file4_pauseandponder.py** + +**file5_surface.py** + diff --git a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/surface-integrals/file5_surface.gif b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/surface-integrals/file5_surface.gif Binary files differnew file mode 100644 index 0000000..27dcac8 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/surface-integrals/file5_surface.gif diff --git a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/surface-integrals/file5_surface.py b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/surface-integrals/file5_surface.py new file mode 100644 index 0000000..3c2e145 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/surface-integrals/file5_surface.py @@ -0,0 +1,237 @@ +from manimlib.imports import * + +class Surf(ThreeDScene): + + CONFIG = { + "axes_config": { + "x_min": 0, + "x_max": 8, + "y_min": 0, + "y_max": 8, + "z_min": 0, + "z_max": 6, + "a":2 ,"b": 6, "c":1 , "d":6, + "axes_shift":-3*OUT + 5*LEFT, + "x_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "y_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "z_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "num_axis_pieces": 1, + }, + "default_graph_style": { + "stroke_width": 2, + "stroke_color": WHITE, + }, + "default_surface_config": { + "fill_opacity": 0.5, + "checkerboard_colors": [LIGHT_GREY], + "stroke_width": 0.5, + "stroke_color": WHITE, + "stroke_opacity": 0.5, + }, + "Func": lambda x,y: 2+y/4+np.sin(x) + } + + + def construct(self): + + self.setup_axes() + self.set_camera_orientation(distance=35, + phi=80 * DEGREES, + theta=-80 * DEGREES, + ) + + fn_text=TextMobject("$S$").set_color(BLUE) + self.add_fixed_in_frame_mobjects(fn_text) + fn_text.to_edge(TOP,buff=MED_SMALL_BUFF) + + R=TextMobject("D").set_color(BLACK).scale(3) + R.move_to(self.axes.input_plane,IN) + self.add(R) + + #get the surface + surface= self.get_surface( + self.axes, lambda x , y: + self.Func(x,y) + ) + surface.set_style( + fill_opacity=0.8, + fill_color=YELLOW, + stroke_width=0.8, + stroke_color=WHITE, + ) + + + self.begin_ambient_camera_rotation(rate=0.05) + self.play(Write(surface)) + # self.play(LaggedStart(ShowCreation(surface))) + + self.get_lines() + # self.play(FadeIn(self.axes.input_plane)) + self.wait(2) + self.stop_ambient_camera_rotation() + self.wait(1) + + def get_surface(self,axes, func, **kwargs): + config = { + "u_min": axes.c, + "u_max": axes.d, + "v_min": axes.a, + "v_max": axes.b, + "resolution": ( + (axes.y_max - axes.y_min) // axes.y_axis.tick_frequency, + (axes.x_max - axes.x_min) // axes.x_axis.tick_frequency, + ), + } + + config.update(self.default_surface_config) + config.update(kwargs) + return ParametricSurface( + lambda x,y : axes.c2p( + x, y, func(x, y) + ), + **config + ) + + def get_lines(self): + axes = self.axes + labels=[axes.x_axis.n2p(axes.a), axes.x_axis.n2p(axes.b), axes.y_axis.n2p(axes.c), + axes.y_axis.n2p(axes.d)] + + + surface_corners=[] + for x,y,z in self.region_corners: + surface_corners.append([x,y,self.Func(x,y)]) + + lines=VGroup() + for start , end in zip(surface_corners, + self.region_corners): + lines.add(self.draw_lines(start,end,"WHITE")) + + for start , end in zip(labels, + self.region_corners): + # lines.add(self.draw_lines(start,end,"BLUE")) + # print (start,end) + pass + self.play(ShowCreation(lines)) + + + def draw_lines(self,start,end,color): + start=self.axes.c2p(*start) + end=self.axes.c2p(*end) + line=DashedLine(start,end,color=color) + + return line + + def get_three_d_axes(self, include_labels=True, include_numbers=True, **kwargs): + config = dict(self.axes_config) + config.update(kwargs) + axes = ThreeDAxes(**config) + axes.set_stroke(width=2) + + if include_numbers: + self.add_axes_numbers(axes) + + if include_labels: + self.add_axes_labels(axes) + + # Adjust axis orientation + axes.x_axis.rotate( + 90 * DEGREES, RIGHT, + about_point=axes.c2p(0, 0, 0), + ) + axes.y_axis.rotate( + 90 * DEGREES, UP, + about_point=axes.c2p(0, 0, 0), + ) + + # Add xy-plane + input_plane = self.get_surface( + axes, lambda x, t: 1e-5 + ) + input_plane.set_style( + fill_opacity=0.5, + fill_color=TEAL, + stroke_width=0, + stroke_color=WHITE, + ) + + axes.input_plane = input_plane + + self.region_corners=[ + input_plane.get_corner(pos) for pos in (DL,DR,UR,UL)] + + return axes + + + def setup_axes(self): + axes = self.get_three_d_axes(include_labels=True) + axes.add(axes.input_plane) + axes.scale(1) + # axes.center() + axes.shift(axes.axes_shift) + + self.add(axes) + self.axes = axes + + def add_axes_numbers(self, axes): + x_axis = axes.x_axis + y_axis = axes.y_axis + tex_vals_x = [ + ("a", axes.a), + ("b", axes.b), + ] + tex_vals_y=[ + ("c", axes.c), + ("d", axes.d) + ] + x_labels = VGroup() + y_labels = VGroup() + for tex, val in tex_vals_x: + label = TexMobject(tex) + label.scale(1) + label.next_to(x_axis.n2p(val), DOWN) + x_labels.add(label) + x_axis.add(x_labels) + x_axis.numbers = x_labels + + for tex, val in tex_vals_y: + label = TexMobject(tex) + label.scale(1.5) + label.next_to(y_axis.n2p(val), LEFT) + label.rotate(90 * DEGREES) + y_labels.add(label) + + y_axis.add(y_labels) + y_axis.numbers = y_labels + + return axes + + def add_axes_labels(self, axes): + x_label = TexMobject("X") + x_label.next_to(axes.x_axis.get_end(), RIGHT) + axes.x_axis.label = x_label + + y_label = TextMobject("Y") + y_label.rotate(90 * DEGREES, OUT) + y_label.next_to(axes.y_axis.get_end(), UP) + axes.y_axis.label = y_label + + z_label = TextMobject("Z") + z_label.rotate(90 * DEGREES, RIGHT) + z_label.next_to(axes.z_axis.get_zenith(), RIGHT) + axes.z_axis.label = z_label + for axis in axes: + axis.add(axis.label) + return axes + ######Code_by_Somnath_Pandit_https://github.com/panditsomnath10016git######### + + diff --git a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/triple-integrals/README.md b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/triple-integrals/README.md index e69de29..2166a79 100644 --- a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/triple-integrals/README.md +++ b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/triple-integrals/README.md @@ -0,0 +1,6 @@ +**file1_3D_crossproduct.py** + +**file2_cylindrical_coordinates.py** + +**file2_spherical_coordinates.py** + diff --git a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/triple-integrals/file2_cylindrical_coordinates.gif b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/triple-integrals/file2_cylindrical_coordinates.gif Binary files differindex 6dc8b17..e913750 100644 --- a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/triple-integrals/file2_cylindrical_coordinates.gif +++ b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/triple-integrals/file2_cylindrical_coordinates.gif diff --git a/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/triple-integrals/file2_spherical_coordinates.gif b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/triple-integrals/file2_spherical_coordinates.gif Binary files differnew file mode 100644 index 0000000..6dc8b17 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/triple-and-surface-integrals/triple-integrals/file2_spherical_coordinates.gif |
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