Update and rename file3_Nondegenerate_Hessian_Matrix.py to file2_Nondegenerate_Hessian_Matrix.py

1 files changed, 158 insertions, 0 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)