SubTopic: Critical Points
- Animations
+ Animations
- file1_Critical_Point_of_a_function
+file1_Critical_Point_of_a_function
![file1_Critical_Point_of_a_function](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.gif?raw=true)
-
-file2_Traces_and_Tangent
+file2_Traces_and_Tangent
- ![file2_Traces_and_Tangent](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif?raw=true)
+![file2_Traces_and_Tangent](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif?raw=true)
-
-file3_Tangent_plane_at_extrema_of_a_function
- ![file3_Tangent_plane_at_extrema_of_a_function](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif?raw=true)
+file3_Tangent_plane_at_extrema_of_a_function
+
+![file3_Tangent_plane_at_extrema_of_a_function](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif?raw=true)
-
-file4_Types_of_critical_points
+file4_Types_of_critical_points
- ![file4_Types_of_critical_points](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif?raw=true)
+![file4_Types_of_critical_points](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif?raw=true)
-
-file5_f(x,y)=(y-x)(1-2x-3y)
+file5_f(x,y)=(y-x)(1-2x-3y)
- ![file5_f(x,y)=(y-x)(1-2x-3y)](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file5_f(x%2Cy)%3D(y-x)(1-2x-3y).gif?raw=true)
+![file5_f(x,y)=(y-x)(1-2x-3y)](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file5_f(x%2Cy)%3D(y-x)(1-2x-3y).gif?raw=true)
-
-
-
-
--
cgit
From 3bf2eb6ac38722218f5a9b3ab89c12629073ed9d Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 01:16:02 +0530
Subject: Update README.md
---
FSF-2020/approximations-and-optimizations/Critical-Points/README.md | 4 ----
1 file changed, 4 deletions(-)
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/Critical-Points/README.md b/FSF-2020/approximations-and-optimizations/Critical-Points/README.md
index 598607b..f0747bb 100644
--- a/FSF-2020/approximations-and-optimizations/Critical-Points/README.md
+++ b/FSF-2020/approximations-and-optimizations/Critical-Points/README.md
@@ -1,9 +1,5 @@
SubTopic: Critical Points
- Animations
-
-
-
file1_Critical_Point_of_a_function
--
cgit
From 14038e183235f0c3c9c8b354e1e16b17728e4bbf Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 01:45:49 +0530
Subject: Create file1_Second_order_partial_derivatives.py
Source code of first animation of the lecture note: The second derivative test---
.../file1_Second_order_partial_derivatives.py | 78 ++++++++++++++++++++++
1 file changed, 78 insertions(+)
create mode 100644 FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.py
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.py b/FSF-2020/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/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)
--
cgit
From 7cfd798434a60d8664b3a0eff04193cfd93b50ab Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 01:49:25 +0530
Subject: Add files via upload
---
.../file1_Second_order_partial_derivatives.gif | Bin 0 -> 3166332 bytes
1 file changed, 0 insertions(+), 0 deletions(-)
create mode 100644 FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.gif
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.gif b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.gif
new file mode 100644
index 0000000..3471e4d
Binary files /dev/null and b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file1_Second_order_partial_derivatives.gif differ
--
cgit
From 14a06e6bf4ed8a5a661555f193ba145fe8121b86 Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 02:04:03 +0530
Subject: Create file4_Contour_Diagram.py
---
.../file4_Contour_Diagram.py | 121 +++++++++++++++++++++
1 file changed, 121 insertions(+)
create mode 100644 FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.py
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.py b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.py
new file mode 100644
index 0000000..618dcbd
--- /dev/null
+++ b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.py
@@ -0,0 +1,121 @@
+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)
+ ]),
+ resolution=(6, 32)).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(f),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)
--
cgit
From 1a3fcb648c8be25ff2a21833c532330be6c5238c Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 02:44:28 +0530
Subject: Update file4_Contour_Diagram.py
---
.../The-Second-Derivative-Test/file4_Contour_Diagram.py | 5 ++---
1 file changed, 2 insertions(+), 3 deletions(-)
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.py b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.py
index 618dcbd..d3084e2 100644
--- a/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.py
+++ b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.py
@@ -50,8 +50,7 @@ class ContourDiagram(ThreeDScene):
np.cos(TAU * v),
np.sin(TAU * v),
2*(1 - 1.5*u)
- ]),
- resolution=(6, 32)).fade(0.5).scale(0.61).shift([0,0,0.4]).set_color(YELLOW)
+ ])).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])
@@ -111,7 +110,7 @@ class ContourDiagram(ThreeDScene):
self.wait(1)
self.play(Write(c_5))
self.wait(1)
- self.play(FadeOut(contour5),FadeOut(axes),FadeOut(f),FadeOut(contour5),FadeOut(contour4),FadeOut(contour3),FadeOut(contour2),FadeOut(contour1))
+ 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)
--
cgit
From 51b69836caaea9743d6d4a3c91a789b3fdc8ceee Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 02:47:35 +0530
Subject: Add files via upload
---
.../file4_Contour_Diagram.gif | Bin 0 -> 1150777 bytes
1 file changed, 0 insertions(+), 0 deletions(-)
create mode 100644 FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.gif
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.gif b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.gif
new file mode 100644
index 0000000..129fedc
Binary files /dev/null and b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file4_Contour_Diagram.gif differ
--
cgit
From 39e28347b3584d3077cefa2b303d6fa6ac802feb Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 03:47:48 +0530
Subject: Create file2_Degenerate_Hessian_Matrix.py
---
.../file2_Degenerate_Hessian_Matrix.py | 52 ++++++++++++++++++++++
1 file changed, 52 insertions(+)
create mode 100644 FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file2_Degenerate_Hessian_Matrix.py
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file2_Degenerate_Hessian_Matrix.py b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file2_Degenerate_Hessian_Matrix.py
new file mode 100644
index 0000000..c1e3516
--- /dev/null
+++ b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file2_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 355abf1850e7884e91f6bcf4fc35272a704b45b7 Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 03:48:41 +0530
Subject: Create file3_Nondegenerate_Hessian_Matrix.py
---
.../file3_Nondegenerate_Hessian_Matrix.py | 145 +++++++++++++++++++++
1 file changed, 145 insertions(+)
create mode 100644 FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file3_Nondegenerate_Hessian_Matrix.py
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file3_Nondegenerate_Hessian_Matrix.py b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file3_Nondegenerate_Hessian_Matrix.py
new file mode 100644
index 0000000..3056842
--- /dev/null
+++ b/FSF-2020/approximations-and-optimizations/The-Second-Derivative-Test/file3_Nondegenerate_Hessian_Matrix.py
@@ -0,0 +1,145 @@
+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 e4de3f4275bf5684fc14873730bfb050b4cf7d17 Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 03:55:58 +0530
Subject: Update and rename file1_TangentPlanes.py to file1_Tangent_Plane.py
---
.../file1_TangentPlanes.py | 39 ----------------
.../file1_Tangent_Plane.py | 52 ++++++++++++++++++++++
2 files changed, 52 insertions(+), 39 deletions(-)
delete mode 100644 FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_TangentPlanes.py
create mode 100644 FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.py
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_TangentPlanes.py b/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_TangentPlanes.py
deleted file mode 100644
index 8e3a19e..0000000
--- a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_TangentPlanes.py
+++ /dev/null
@@ -1,39 +0,0 @@
-from manimlib.imports import*
-
-class TangenttoSurface(ThreeDScene):
- def construct(self):
- axes = ThreeDAxes()
-
- #parabola: -x**2-y**2
- p = 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_C,TEAL_D],
- resolution=(20, 20)).scale(1).shift(1*RIGHT+2*UP)
- self.set_camera_orientation(phi = 35 * DEGREES,theta = -40 * DEGREES )
-
- r = Rectangle(side_length=2,side_breadth= 1, fill_color=PURPLE, fill_opacity=0.2).shift(ORIGIN-1+3*UP+2*RIGHT).scale(0.7) #---tangent plane along x axis
-
- r_text = TextMobject("Tangent Plane along $x$ axis",color = '#FFE4E1').scale(0.6).to_corner(UL)
- r2_text = TextMobject("Tangent Plane along $y$ axis",color = '#FFE4E1').scale(0.6).to_corner(UL)
-
- a = Arrow(color = '#FFFFF0').shift(ORIGIN-1+3*UP+4*RIGHT).scale(0.5)
- a2 = Arrow(color = '#FFFFF0').shift(ORIGIN+0.5+3*UP+RIGHT).scale(0.5)
- a2.rotate(1.571) #----1.571 radian = 90 degrees
-
- self.add(axes)
- self.play(Write(p))
- self.begin_ambient_camera_rotation(rate = 0.1)
- self.add_fixed_in_frame_mobjects(r_text)
- self.play(ShowCreation(r))
- self.play(ShowCreation(a))
- self.wait(1)
- self.play(FadeOut(r),FadeOut(a),FadeOut(r_text))
-
- r.rotate(1.571) #---tangent plane along y axis
- self.play(ShowCreation(r))
- self.play(ShowCreation(a2))
- self.add_fixed_in_frame_mobjects(r2_text)
- self.wait(2)
diff --git a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.py b/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.py
new file mode 100644
index 0000000..3632d7e
--- /dev/null
+++ b/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.py
@@ -0,0 +1,52 @@
+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)
+
+
--
cgit
From e59375a9f91c17376c690fd59f86cfe0490018d0 Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 03:56:17 +0530
Subject: Update file1_Tangent_Plane.py
---
.../Tangent-Plane-Approximations/file1_Tangent_Plane.py | 2 --
1 file changed, 2 deletions(-)
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.py b/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.py
index 3632d7e..4c17f90 100644
--- a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.py
+++ b/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.py
@@ -48,5 +48,3 @@ class firstScene(ThreeDScene):
self.wait(2)
self.play(ReplacementTransform(f,s),ReplacementTransform(d,d2),ReplacementTransform(r,r2))
self.wait(2)
-
-
--
cgit
From 6a369492d9bfceb37b27856c082d218f50dbcb87 Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 03:56:37 +0530
Subject: Delete file1_TangentPlanes.mp4
---
.../file1_TangentPlanes.mp4 | Bin 1483459 -> 0 bytes
1 file changed, 0 insertions(+), 0 deletions(-)
delete mode 100644 FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_TangentPlanes.mp4
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_TangentPlanes.mp4 b/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_TangentPlanes.mp4
deleted file mode 100644
index 6a3ba6d..0000000
Binary files a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_TangentPlanes.mp4 and /dev/null differ
--
cgit
From 4d945dced9720f944215a39d10f78206334f2f61 Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 03:57:12 +0530
Subject: Delete file2_Tangent_plane_approximation_visualization.mp4
---
...file2_Tangent_plane_approximation_visualization.mp4 | Bin 336179 -> 0 bytes
1 file changed, 0 insertions(+), 0 deletions(-)
delete mode 100644 FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file2_Tangent_plane_approximation_visualization.mp4
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file2_Tangent_plane_approximation_visualization.mp4 b/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file2_Tangent_plane_approximation_visualization.mp4
deleted file mode 100644
index bc3448b..0000000
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--
cgit
From dd56199fd1aa1e1dc74d7d05280093cc5f5e66b9 Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 03:57:23 +0530
Subject: Delete file3_non_differentiable_function.mp4
---
.../file3_non_differentiable_function.mp4 | Bin 424364 -> 0 bytes
1 file changed, 0 insertions(+), 0 deletions(-)
delete mode 100644 FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file3_non_differentiable_function.mp4
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file3_non_differentiable_function.mp4 b/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file3_non_differentiable_function.mp4
deleted file mode 100644
index 7417a87..0000000
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--
cgit
From e37b2e1ad48269d1fa32c402dbf85331b6490376 Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 03:57:34 +0530
Subject: Delete file4_Tangent plane_at_critical_points.mp4
---
.../file4_Tangent plane_at_critical_points.mp4 | Bin 3101810 -> 0 bytes
1 file changed, 0 insertions(+), 0 deletions(-)
delete mode 100644 FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file4_Tangent plane_at_critical_points.mp4
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file4_Tangent plane_at_critical_points.mp4 b/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file4_Tangent plane_at_critical_points.mp4
deleted file mode 100644
index 2c64f38..0000000
Binary files a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file4_Tangent plane_at_critical_points.mp4 and /dev/null differ
--
cgit
From ada683cd4827f2b996998a10ed5c967b7945d530 Mon Sep 17 00:00:00 2001
From: Vaishnavi
Date: Wed, 24 Jun 2020 03:58:34 +0530
Subject: Add files via upload
---
.../file1_Tangent_Plane.gif | Bin 0 -> 827096 bytes
1 file changed, 0 insertions(+), 0 deletions(-)
create mode 100644 FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.gif
(limited to 'FSF-2020')
diff --git a/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.gif b/FSF-2020/approximations-and-optimizations/Tangent-Plane-Approximations/file1_Tangent_Plane.gif
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index 0000000..2b8bf5f
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--
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