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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) |
