From 66e1207623862b92d68ccff098705194e05516de Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 23:32:46 +0530 Subject: Update and rename file3_Nondegenerate_Hessian_Matrix.py to file2_Nondegenerate_Hessian_Matrix.py --- .../file2_Nondegenerate_Hessian_Matrix.py | 158 +++++++++++++++++++++ .../file3_Nondegenerate_Hessian_Matrix.py | 145 ------------------- 2 files changed, 158 insertions(+), 145 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Nondegenerate_Hessian_Matrix.py delete mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Nondegenerate_Hessian_Matrix.py (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test') 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 $").to_corner(UL).scale(0.7) + + #---- function p(x,y) + p_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2-v**2 + ]),u_min = -1, u_max = 1, v_min = -1, v_max =1).set_color(TEAL).shift([0,0,0]).scale(2) + + p_text= TextMobject("surface of the function").to_corner(UL).scale(0.8) + + d = Dot(color = "#800000").shift([0,0,0]) #---- critical point + + self.set_camera_orientation(phi = 80*DEGREES, theta = 60*DEGREES) + self.add_fixed_in_frame_mobjects(h_text) + self.wait(1) + self.play(FadeOut(h_text)) + self.wait(1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.wait(1) + self.play(Write(hessian_surface)) + self.play(Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(det_text) + self.wait(2) + self.play(FadeOut(det_text),ReplacementTransform(hessian_surface,p_surface)) + self.add_fixed_in_frame_mobjects(p_text) + self.wait(2) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Nondegenerate_Hessian_Matrix.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Nondegenerate_Hessian_Matrix.py deleted file mode 100644 index 3056842..0000000 --- a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Nondegenerate_Hessian_Matrix.py +++ /dev/null @@ -1,145 +0,0 @@ -from manimlib.imports import* - -class firstScene(Scene): - def construct(self): - - e_text = TextMobject("Case 3: One positive and one negative eigenvalue", color = YELLOW).scale(1).shift(3*UP+1*LEFT) - f_text = TextMobject("$f(x,y) = x^2-2y^2-2x$").scale(0.8).next_to(e_text).shift(6*LEFT+DOWN) - c_text = TextMobject("Critical Point: $(1,0)$").scale(0.8).next_to(f_text).shift(DOWN+4*LEFT) - d_text = TextMobject("\\begin{equation*} D_2(1,0)= \\begin{vmatrix} 2 \\space & 0\\space \\\\ 0 & -4 \\end{vmatrix} \\end{equation*}",color = GREEN).scale(0.9) - - t_text = TextMobject("$D_2 = -8<0$ (Saddle Point)", color = BLUE).scale(0.9).shift(2*DOWN) - - self.play(ShowCreation(e_text)) - self.wait(1) - self.play(ShowCreation(f_text)) - self.wait(1) - self.play(ShowCreation(c_text)) - self.wait(1) - self.play(ShowCreation(d_text)) - self.wait(1) - self.play(ShowCreation(t_text)) - self.wait(2) - -class SaddlePoint(ThreeDScene): - def construct(self): - axes = ThreeDAxes() - f = ParametricSurface( - lambda u, v: np.array([ - u, - v, - u**2-2*v**2-2*u - ]),v_min=-1,v_max=1,u_min=-1,u_max=1,checkerboard_colors=[RED_C,PURPLE_D,YELLOW_E], - resolution=(20, 20)).scale(1) - - self.set_camera_orientation(phi=35 * DEGREES,theta=80*DEGREES) - self.begin_ambient_camera_rotation(rate=0.4) - - f_text = TextMobject("$f(x,y) = x^2-2y^2-2x$",color = GREEN).shift(2*DOWN+2*RIGHT).scale(0.8) - self.add_fixed_in_frame_mobjects(f_text) - self.add(axes) - self.play(Write(f)) - self.wait(3) - - -class secondScene(Scene): - def construct(self): - - h_text = TextMobject("NonDegenerate Hessian Matrix", color = GREEN).scale(1).shift(UP) - e_text = TextMobject("Case 1: Two positive eigenvalues", color = PINK).scale(1).shift(3*UP+2*LEFT) - f_text = TextMobject("$f(x,y) = 2x^2+3y^2-2yx$",color = TEAL).scale(0.8).next_to(e_text).shift(6*LEFT+DOWN) - c_text = TextMobject("Critical Point: $(0,0)$",color = TEAL).scale(0.8).next_to(f_text).shift(DOWN+4.5*LEFT) - d_text = TextMobject("\\begin{equation*} D_2(0,0)= \\begin{vmatrix} 4 \\space & -2\\space \\\\ -2 & 6 \\end{vmatrix} \\end{equation*}",color = PINK).scale(0.9) - - t_text = TextMobject("$D_2 = 20>0$ (Relative Maxima or Relative Minima)", color = YELLOW).scale(0.9).shift(1*DOWN) - tm_text = TextMobject("$D_1 = \\frac{\\partial^2 f}{\\partial x^2} =4 >0$ (Relative Minima)", color = YELLOW).scale(0.9).shift(2*DOWN) - - - self.play(ShowCreation(h_text)) - self.wait(1) - self.play(FadeOut(h_text)) - self.wait(1) - self.play(ShowCreation(e_text)) - self.wait(1) - self.play(ShowCreation(f_text)) - self.wait(1) - self.play(ShowCreation(c_text)) - self.wait(1) - self.play(ShowCreation(d_text)) - self.wait(1) - self.play(ShowCreation(t_text)) - self.wait(1) - self.play(ShowCreation(tm_text)) - self.wait(2) - self.play(FadeOut(e_text),FadeOut(f_text),FadeOut(c_text),FadeOut(d_text),FadeOut(t_text),FadeOut(tm_text)) - -class Minima(ThreeDScene): - def construct(self): - axes = ThreeDAxes() - f = ParametricSurface( - lambda u, v: np.array([ - u, - v, - 2*u**2+3*v**2-2*v*u - ]),v_min=-1,v_max=1,u_min=-1,u_max=1,checkerboard_colors=[BLUE_C,YELLOW_D,GREEN_E], - resolution=(20, 20)).scale(1) - - self.set_camera_orientation(phi=10 * DEGREES,theta=90*DEGREES) - self.begin_ambient_camera_rotation(rate=0.2) - - f_text = TextMobject("$f(x,y) = 2x^2+3y^2-2yx$",color = PURPLE).shift(2*DOWN+3*RIGHT).scale(0.8) - self.add_fixed_in_frame_mobjects(f_text) - self.add(axes) - self.play(Write(f)) - self.wait(2) - - -class thirdScene(Scene): - def construct(self): - - - e_text = TextMobject("Case 2: Two negative eigenvalues", color = RED).scale(1).shift(3*UP+2*LEFT) - f_text = TextMobject("$f(x,y) = -x^2-4y^2$",color = BLUE).scale(0.8).next_to(e_text).shift(6*LEFT+DOWN) - c_text = TextMobject("Critical Point: $(0,0)$",color = BLUE).scale(0.8).next_to(f_text).shift(DOWN+3.8*LEFT) - d_text = TextMobject("\\begin{equation*} D_2(0,0)= \\begin{vmatrix} -2 \\space & 0\\space \\\\ 0 & -8 \\end{vmatrix} \\end{equation*}",color = TEAL).scale(0.9) - - t_text = TextMobject("$D_2 = 16>0$ (Relative Maxima or Relative Minima)" ).scale(0.9).shift(1*DOWN) - tm_text = TextMobject("$D_1 = \\frac{\\partial^2 f}{\\partial x^2} =-2 <0$ (Relative Maxima)").scale(0.9).shift(2*DOWN) - - - self.play(ShowCreation(e_text)) - self.wait(1) - self.play(ShowCreation(f_text)) - self.wait(1) - self.play(ShowCreation(c_text)) - self.wait(1) - self.play(ShowCreation(d_text)) - self.wait(1) - self.play(ShowCreation(t_text)) - self.wait(1) - self.play(ShowCreation(tm_text)) - self.wait(2) - self.play(FadeOut(e_text),FadeOut(f_text),FadeOut(c_text),FadeOut(d_text),FadeOut(t_text),FadeOut(tm_text)) - - -class Maxima(ThreeDScene): - def construct(self): - axes = ThreeDAxes() - f = ParametricSurface( - lambda u, v: np.array([ - u, - v, - -u**2-4*v**2 - ]),v_min=-1,v_max=1,u_min=-1,u_max=1,checkerboard_colors=[BLUE_C,PURPLE_D,TEAL_E], - resolution=(20, 20)).scale(1) - - self.set_camera_orientation(phi=75 * DEGREES) - self.begin_ambient_camera_rotation(rate=0.4) - - f_text = TextMobject("$f(x,y) = -x^2-4y^2$",color = YELLOW).shift(2*DOWN+3*RIGHT).scale(0.8) - self.add_fixed_in_frame_mobjects(f_text) - self.add(axes) - self.play(Write(f)) - self.wait(1) - self.move_camera(phi=30*DEGREES,theta=45*DEGREES,run_time=5) - self.wait(2) -- cgit From 167452dd7e68203a39990f5af4ea99404078d958 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 23:33:12 +0530 Subject: Rename file2_Degenerate_Hessian_Matrix.py to file3_Degenerate_Hessian_Matrix.py --- .../file2_Degenerate_Hessian_Matrix.py | 52 ---------------------- .../file3_Degenerate_Hessian_Matrix.py | 52 ++++++++++++++++++++++ 2 files changed, 52 insertions(+), 52 deletions(-) delete mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Degenerate_Hessian_Matrix.py create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Degenerate_Hessian_Matrix.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Degenerate_Hessian_Matrix.py deleted file mode 100644 index c1e3516..0000000 --- a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Degenerate_Hessian_Matrix.py +++ /dev/null @@ -1,52 +0,0 @@ -from manimlib.imports import* - -class firstscene(Scene): - def construct(self): - - h_text = TextMobject("Degenerate Hessian Matrix", color = RED).scale(1).shift(UP) - - - f_text = TextMobject("$f(x,y) = 2x^3+y^3$", color = TEAL).scale(1).to_corner(UL) - c_text = TextMobject("Critical Point: $(0,0)$", color = TEAL).scale(1).next_to(f_text).shift(DOWN+4.3*LEFT) - m_text = TextMobject("\\begin{equation*} D_2(x,y)= \\begin{vmatrix} 12x\\space & 0\\space \\\\ 0 & 6y \\end{vmatrix} \\end{equation*}",color = YELLOW) - d_text = TextMobject("\\begin{equation*} D_2(0,0)= \\begin{vmatrix} 0 \\space & 0\\space \\\\ 0 & 0 \\end{vmatrix} \\end{equation*}",color = PURPLE) - - - t_text = TextMobject("$D_2 = 0$(Inconclusive)", color = TEAL).scale(1).shift(2*DOWN) - - self.play(ShowCreation(h_text)) - self.wait(1) - self.play(FadeOut(h_text)) - self.wait(1) - self.play(ShowCreation(f_text)) - self.wait(1) - self.play(ShowCreation(c_text)) - self.wait(1) - self.play(ShowCreation(m_text)) - self.wait(2) - self.play(ReplacementTransform(m_text,d_text)) - self.wait(1) - self.play(ShowCreation(t_text)) - self.wait(2) - - -class SecondScene(ThreeDScene): - def construct(self): - axes = ThreeDAxes() - - f = ParametricSurface( - lambda u, v: np.array([ - u, - v, - (2*u**3)+v**3 - ]),v_min=-1,v_max=1,u_min=-1,u_max=1,checkerboard_colors=[TEAL_C,YELLOW_D,BLUE_E], - resolution=(20, 20)).scale(1) - - self.set_camera_orientation(phi=25 * DEGREES,theta = 80*DEGREES) - self.begin_ambient_camera_rotation(rate=0.1) - - f_text = TextMobject("$f(x,y) = 2x^3+y^3$",color = ORANGE).shift(2*DOWN+2*RIGHT).scale(0.8) - self.add_fixed_in_frame_mobjects(f_text) - self.add(axes) - self.play(Write(f)) - self.wait(2) 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..c1e3516 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py @@ -0,0 +1,52 @@ +from manimlib.imports import* + +class firstscene(Scene): + def construct(self): + + h_text = TextMobject("Degenerate Hessian Matrix", color = RED).scale(1).shift(UP) + + + f_text = TextMobject("$f(x,y) = 2x^3+y^3$", color = TEAL).scale(1).to_corner(UL) + c_text = TextMobject("Critical Point: $(0,0)$", color = TEAL).scale(1).next_to(f_text).shift(DOWN+4.3*LEFT) + m_text = TextMobject("\\begin{equation*} D_2(x,y)= \\begin{vmatrix} 12x\\space & 0\\space \\\\ 0 & 6y \\end{vmatrix} \\end{equation*}",color = YELLOW) + d_text = TextMobject("\\begin{equation*} D_2(0,0)= \\begin{vmatrix} 0 \\space & 0\\space \\\\ 0 & 0 \\end{vmatrix} \\end{equation*}",color = PURPLE) + + + t_text = TextMobject("$D_2 = 0$(Inconclusive)", color = TEAL).scale(1).shift(2*DOWN) + + self.play(ShowCreation(h_text)) + self.wait(1) + self.play(FadeOut(h_text)) + self.wait(1) + self.play(ShowCreation(f_text)) + self.wait(1) + self.play(ShowCreation(c_text)) + self.wait(1) + self.play(ShowCreation(m_text)) + self.wait(2) + self.play(ReplacementTransform(m_text,d_text)) + self.wait(1) + self.play(ShowCreation(t_text)) + self.wait(2) + + +class SecondScene(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + + f = ParametricSurface( + lambda u, v: np.array([ + u, + v, + (2*u**3)+v**3 + ]),v_min=-1,v_max=1,u_min=-1,u_max=1,checkerboard_colors=[TEAL_C,YELLOW_D,BLUE_E], + resolution=(20, 20)).scale(1) + + self.set_camera_orientation(phi=25 * DEGREES,theta = 80*DEGREES) + self.begin_ambient_camera_rotation(rate=0.1) + + f_text = TextMobject("$f(x,y) = 2x^3+y^3$",color = ORANGE).shift(2*DOWN+2*RIGHT).scale(0.8) + self.add_fixed_in_frame_mobjects(f_text) + self.add(axes) + self.play(Write(f)) + self.wait(2) -- cgit From 1a08a9dcea173dcd180078a8032c2bb8853bcda2 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 23:33:25 +0530 Subject: Delete file2_Degenerate_Hessian_Matrix.gif --- .../file2_Degenerate_Hessian_Matrix.gif | Bin 3202838 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Degenerate_Hessian_Matrix.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Degenerate_Hessian_Matrix.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Degenerate_Hessian_Matrix.gif deleted file mode 100644 index d49cdd5..0000000 Binary files a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Degenerate_Hessian_Matrix.gif and /dev/null differ -- cgit From a9afb294a078ea365a1a42d714deb2a5bf5312f8 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 23:34:19 +0530 Subject: Add files via upload --- .../file2_Nondegenerate_Hessian_Matrix.gif | Bin 0 -> 2047897 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Nondegenerate_Hessian_Matrix.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test') 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 new file mode 100644 index 0000000..0d58b4f Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Nondegenerate_Hessian_Matrix.gif differ -- cgit From 2008bd765735cdef19a451940bfb6d0b3958c883 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Sat, 27 Jun 2020 02:26:47 +0530 Subject: Update file3_Degenerate_Hessian_Matrix.py --- .../file3_Degenerate_Hessian_Matrix.py | 83 ++++++++++++---------- 1 file changed, 44 insertions(+), 39 deletions(-) (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test') 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 index c1e3516..4f76daf 100644 --- 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 @@ -1,52 +1,57 @@ from manimlib.imports import* - -class firstscene(Scene): +import math as m + +class DegenerateHessian(ThreeDScene): def construct(self): - h_text = TextMobject("Degenerate Hessian Matrix", color = RED).scale(1).shift(UP) - - - f_text = TextMobject("$f(x,y) = 2x^3+y^3$", color = TEAL).scale(1).to_corner(UL) - c_text = TextMobject("Critical Point: $(0,0)$", color = TEAL).scale(1).next_to(f_text).shift(DOWN+4.3*LEFT) - m_text = TextMobject("\\begin{equation*} D_2(x,y)= \\begin{vmatrix} 12x\\space & 0\\space \\\\ 0 & 6y \\end{vmatrix} \\end{equation*}",color = YELLOW) - d_text = TextMobject("\\begin{equation*} D_2(0,0)= \\begin{vmatrix} 0 \\space & 0\\space \\\\ 0 & 0 \\end{vmatrix} \\end{equation*}",color = PURPLE) + 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) - t_text = TextMobject("$D_2 = 0$(Inconclusive)", color = TEAL).scale(1).shift(2*DOWN) + 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 - self.play(ShowCreation(h_text)) + #---- function f(x,y) + f_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -4*u**3-v**3 + ]),u_min = -0.8, u_max = 0.8, v_min = -0.8, v_max = 0.8).set_color(TEAL).shift([0,1,0]).scale(1.3) + + #---- function f(x,y) + zoom_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -4*u**3-v**3 + ]),u_min = -0.8, u_max = 0.8, v_min = -0.8, v_max = 0.8).set_color(TEAL).shift([0,1,0]).scale(2.5) + + f_text= TextMobject("surface of the function").to_corner(UL).scale(0.5) + + d = Dot(color = "#800000").shift([0,1,0]) #---- critical point + d2 = Dot(color = "#800000").shift([0,0.7,0]) #---- critical point + plane = Rectangle(color = YELLOW,fill_opacity= 0.3).shift([0,0.6,0]).rotate(m.radians(90)).scale(0.4) + + 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.play(ShowCreation(f_text)) - self.wait(1) - self.play(ShowCreation(c_text)) + 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(ShowCreation(m_text)) + self.play(Write(d)) + self.wait(1) + self.play(ReplacementTransform(f_surface,zoom_surface),ReplacementTransform(d,d2)) self.wait(2) - self.play(ReplacementTransform(m_text,d_text)) + self.play(Write(plane)) self.wait(1) - self.play(ShowCreation(t_text)) - self.wait(2) - - -class SecondScene(ThreeDScene): - def construct(self): - axes = ThreeDAxes() - f = ParametricSurface( - lambda u, v: np.array([ - u, - v, - (2*u**3)+v**3 - ]),v_min=-1,v_max=1,u_min=-1,u_max=1,checkerboard_colors=[TEAL_C,YELLOW_D,BLUE_E], - resolution=(20, 20)).scale(1) - - self.set_camera_orientation(phi=25 * DEGREES,theta = 80*DEGREES) - self.begin_ambient_camera_rotation(rate=0.1) - - f_text = TextMobject("$f(x,y) = 2x^3+y^3$",color = ORANGE).shift(2*DOWN+2*RIGHT).scale(0.8) - self.add_fixed_in_frame_mobjects(f_text) - self.add(axes) - self.play(Write(f)) - self.wait(2) -- cgit From 4280330114ad16782a6b584c26adf78c1820d298 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Sat, 27 Jun 2020 02:27:07 +0530 Subject: Update file3_Degenerate_Hessian_Matrix.py --- .../The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py | 1 - 1 file changed, 1 deletion(-) (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test') 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 index 4f76daf..732ef4f 100644 --- 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 @@ -54,4 +54,3 @@ class DegenerateHessian(ThreeDScene): self.wait(2) self.play(Write(plane)) self.wait(1) - -- cgit From 38eac13cd8fc554d4bf10c5cc0a6c2dbfb31ca7b Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Sat, 27 Jun 2020 02:27:47 +0530 Subject: Delete file3_Nondegenerate_Hessian_Matrix.gif --- .../file3_Nondegenerate_Hessian_Matrix.gif | Bin 8724439 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Nondegenerate_Hessian_Matrix.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Nondegenerate_Hessian_Matrix.gif 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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 new file mode 100644 index 0000000..36fd25c Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.gif differ -- cgit From 84569f5fcc0195a488e055b5566225235d93011f Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Sat, 27 Jun 2020 03:03:52 +0530 Subject: Create README.md --- .../The-Second-Derivative-Test/README.md | 21 +++++++++++++++++++++ 1 file changed, 21 insertions(+) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/README.md (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test') 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..e9cc918 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/README.md @@ -0,0 +1,21 @@ +

SubTopic: The Second Derivative Test

+

+ +file1_Second_order_partial_derivatives + +![file1_Second_order_partial_derivatives](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.gif?raw=true) +

+

+ +file2_Nondegenerate_Hessian_Matrix + +![file2_Nondegenerate_Hessian_Matrix](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file2_Nondegenerate_Hessian_Matrix.gif?raw=true) +

+

+ +file3_Degenerate_Hessian_Matrix + +![file3_Degenerate_Hessian_Matrix](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.gif?raw=true) +

+

+ -- cgit