Create file3_Nondegenerate_Hessian_Matrix.py

1 files changed, 145 insertions, 0 deletions

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)