diff options
Diffstat (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py')
| -rw-r--r-- | FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py | 83 |
1 files changed, 44 insertions, 39 deletions
diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py index c1e3516..4f76daf 100644 --- a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/The-Second-Derivative-Test/file3_Degenerate_Hessian_Matrix.py @@ -1,52 +1,57 @@ from manimlib.imports import* - -class firstscene(Scene): +import math as m + +class DegenerateHessian(ThreeDScene): 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) + heading = TextMobject("Degenerate Hessian Matrix",color = BLUE) + h_text = TextMobject("For $det \\hspace{1mm} H = 0$, the surface of the function at the critical point would be flat.").scale(0.7) - t_text = TextMobject("$D_2 = 0$(Inconclusive)", color = TEAL).scale(1).shift(2*DOWN) + 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 - self.play(ShowCreation(h_text)) + #---- function f(x,y) + f_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -4*u**3-v**3 + ]),u_min = -0.8, u_max = 0.8, v_min = -0.8, v_max = 0.8).set_color(TEAL).shift([0,1,0]).scale(1.3) + + #---- function f(x,y) + zoom_surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -4*u**3-v**3 + ]),u_min = -0.8, u_max = 0.8, v_min = -0.8, v_max = 0.8).set_color(TEAL).shift([0,1,0]).scale(2.5) + + f_text= TextMobject("surface of the function").to_corner(UL).scale(0.5) + + d = Dot(color = "#800000").shift([0,1,0]) #---- critical point + d2 = Dot(color = "#800000").shift([0,0.7,0]) #---- critical point + plane = Rectangle(color = YELLOW,fill_opacity= 0.3).shift([0,0.6,0]).rotate(m.radians(90)).scale(0.4) + + self.set_camera_orientation(phi = 70*DEGREES, theta = 45*DEGREES) + self.add_fixed_in_frame_mobjects(heading) self.wait(1) + self.play(FadeOut(heading)) + self.add_fixed_in_frame_mobjects(h_text) + self.wait(2) self.play(FadeOut(h_text)) self.wait(1) - self.play(ShowCreation(f_text)) - self.wait(1) - self.play(ShowCreation(c_text)) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(f_surface)) + self.add_fixed_in_frame_mobjects(f_text) self.wait(1) - self.play(ShowCreation(m_text)) + self.play(Write(d)) + self.wait(1) + self.play(ReplacementTransform(f_surface,zoom_surface),ReplacementTransform(d,d2)) self.wait(2) - self.play(ReplacementTransform(m_text,d_text)) + self.play(Write(plane)) 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) |