1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
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)
|
