Update and rename file3_Nondegenerate_Hessian_Matrix.py to file2_Nondegenerate_Hessian_Matrix.py

2 files changed, 158 insertions, 145 deletions

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)