Create file1_Second_order_partial_derivatives.py

Source code of first animation of the lecture note: The second derivative test

1 files changed, 78 insertions, 0 deletions

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)