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divergencia_Rn-R_infinito.py
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divergencia_Rn-R_infinito.py
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from manimlib.imports import *
# Cuando el lim es igual a infinito
class superficie3(ParametricSurface):
def __init__(self, **kwargs):
kwargs = {
"u_min": -3,
"u_max": 3,
"v_min": -3,
"v_max": 3,
"checkerboard_colors": [BLUE_E]
}
ParametricSurface.__init__(self, self.func, **kwargs)
def func(self, x, y):
return np.array([x,y,(x*x)+(y*y)-1])
class superficie4(ParametricSurface):
def __init__(self, **kwargs):
kwargs = {
"u_min": 0.1,
"u_max": 5,
"v_min": 0.1,
"v_max": 5,
"checkerboard_colors": [GREEN_B]
}
ParametricSurface.__init__(self, self.func, **kwargs)
def func(self, x, y):
return np.array([x,y,1+(1/((x*x)+(y*y)))])
class LimitesRnaR (ThreeDScene):
def construct(self):
titulo=TextMobject('''Divergencia a Infinito de Funciones \n
de $\\mathbb{R}^{n}\\rightarrow\\mathbb{R}$ en Infinito''').scale(1.5)
text1=TextMobject('''Sea $f:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}$''').move_to(2*UP)
text2=TexMobject(r"lím_{\vec{x}\rightarrow\infty}f(\vec{x})=\infty^{+} \leftrightarrow\forall\ M\in\mathbb{R}").move_to(0.8*UP)
text3=TextMobject('''$\\exists\\delta>0$ tal que si $\\vec{x}\\in B^{c}_{\\delta}(\\vec{0})$ ''' ).move_to(0.5*DOWN)
text4=TexMobject(r'''\implies f(\vec{x})>M''').move_to(1.6*DOWN)
text5=TextMobject("Veamos el siguiente ejemplo para aterrizar lo anterior.")
text6=TextMobject('''Tomemos el paraboloide:\n
$f(x,y)=y^{2}+x^{2}-1$''')
self.play(Write(titulo))
self.wait(5.3)
self.play(FadeOut(titulo))
self.play(Write(text1))
self.play(Write(text2))
self.play(Write(text3))
self.play(Write(text4))
self.wait(8)
self.play(FadeOut(text1),FadeOut(text2),FadeOut(text3),FadeOut(text4))
self.play(Write(text5))
self.wait(5)
self.play(FadeOut(text5))
self.play(Write(text6))
self.wait(3.8)
self.play(FadeOut(text6))
self.custom_method()
def custom_method(self):
axes=ThreeDAxes()
superficie=superficie3()
superficie.set_opacity(0.8)
text1=TexMobject(r'''f(x,y)=y^{2}+x^{2}-1''')
text1.to_corner(UL)
text2=TextMobject('''Tomemos M=0''')
text2.to_corner(UL)
text3=TextMobject('''Tomamos $\\delta$''')
text3.to_corner(UL)
text4=TextMobject('''Veremos que la imagen de los puntos que no \n
están en la bola, son mayor a M''')
text4.to_corner(UL)
text5=TextMobject('''Podemos realizar lo mismo con cualquier M$\\in\\mathbb{R}$''')
text5.to_corner(UL)
text6=TextMobject('''Por lo cual notaremos que la función diverge a $+\\infty$ \n
cuando $\\vec{x}\\rightarrow\\infty$.''')
text6.to_corner(UL)
# text7=TextMobject('''¿Se te ocurre como modificar la definición \n
# cuando la función diverge a $\\infty^{-}$''')
M=0
r=M+1.4
#cilindro = ParametricSurface(
# lambda u, v: np.array([
# r*np.cos(TAU * v),
# r*np.sin(TAU * v),
# 2*u
# ]),
# resolution=(6, 32)).fade(0.1).set_opacity(0.4)
# cilindro.set_color(RED_C).move_to(M*IN)
#cilindro.set_opacity(0.4)
M1=-0.5
r1=M1+1.5
#cilindro1 = ParametricSurface(
# lambda u, v: np.array([
# r1*np.cos(TAU * v),
# r1*np.sin(TAU * v),
# 4*u
# ]),
# resolution=(6, 32)).fade(0.1).set_opacity(0.4)
#cilindro1.set_color(RED_C).move_to((M1/2)*IN)
# cilindro2.set_opacity(0.4)
#bola1=Circle(radius=r1,color=RED,color_opacity=1).move_to(M1*OUT)
bola1=Circle(radius=r1,color=RED,color_opacity=1)
#plano1=Rectangle(height=3, width=5,color=PURPLE_C,fill_color=PURPLE_C,fill_opacity=0.4,
# color_opacity=0.4 ).move_to(M*OUT)
bola=Circle(radius=r,color=RED,color_opacity=1).move_to(M*OUT)
linealabel=TexMobject(r'''\delta''').next_to(bola,RIGHT,buff=0.5).set_color(RED_C).rotate(PI/2,axis=RIGHT).scale(2)
linea=Line((0,0,0),(r,0,0),stroke_width=3,color=RED_C)
def puntosEnSuperficie(rad,lim,num):
puntosDom = []
puntosSur = []
for i in range(num):
azar = np.random.uniform(-lim,lim, (1,2))[0]
if ((rad < np.sqrt(azar[0]**2 + azar[1]**2)) and not (azar[0]<0 and azar[1]>0)):
puntosDom.append(Dot(np.array([azar[0], azar[1],0]), color = PURPLE))
puntosSur.append(Dot(superficie.func(azar[0], azar[1]), color = RED))
return puntosDom, puntosSur
puntosD1, puntosS1 = puntosEnSuperficie(r, 3, 6000)
puntosD2, puntosS2 = puntosEnSuperficie(r1, r, 3000)
GPuntosD1 = VGroup(*puntosD1)
GPuntosS1 = VGroup(*puntosS1)
GPuntosD2 = VGroup(*puntosD2)
GPuntosS2 = VGroup(*puntosS2)
###Animacion
self.set_camera_orientation(0.8*np.pi/2, -0.25*np.pi,distance=15)
self.begin_ambient_camera_rotation(rate=0.001)
self.play(ShowCreation(axes))
self.add_fixed_in_frame_mobjects(text1)
self.play(Write(text1))
self.play(ShowCreation(superficie))
self.wait(2)
self.play(FadeOut(text1))
self.add_fixed_in_frame_mobjects(text2)
self.play(Write(text2))
#self.play(ShowCreation(plano1))
self.wait(2.75)
self.play(FadeOut(text2))
self.add_fixed_in_frame_mobjects(text3)
self.play(Write(text3))
self.play(ShowCreation(linea))
self.play(Write(linealabel))
self.play(ShowCreation(bola))
self.play(FadeOut(linea),FadeOut(linealabel))
self.play(FadeOut(text3))
self.add_fixed_in_frame_mobjects(text4)
self.play(Write(text4))
self.play(FadeIn(GPuntosD1))
self.play(FadeIn(GPuntosS1))
#self.play(ShowCreation(cilindro))
self.wait(8.3)
self.play(FadeOut(text4))
self.add_fixed_in_frame_mobjects(text5)
self.play(Write(text5))
##self.play(plano1.shift,M1*OUT,runtime=1.5)
self.play(ReplacementTransform(bola,bola1))
self.wait()
self.play(FadeIn(GPuntosD2))
self.play(FadeIn(GPuntosS2))
#self.play(ReplacementTransform(cilindro,cilindro1))
self.wait(4.6)
self.play(FadeOut(text5))
self.add_fixed_in_frame_mobjects(text6)
self.play(Write(text6))
self.wait(6.5)
self.play(FadeOut(axes),FadeOut(text6),FadeOut(superficie),FadeOut(bola1),
FadeOut(GPuntosD1),FadeOut(GPuntosS1),FadeOut(GPuntosD2),FadeOut(GPuntosS2))