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from manimlib.imports import *
class LineIntegrationProcess(GraphScene):
CONFIG = {
"x_min" : -0,
"x_max" : 1,
"y_min" : -0,
"y_max" : 1,
"axes_color":WHITE,
"graph_origin": ORIGIN+6.3*LEFT+3*DOWN,
"x_axis_width": 5.5,
"y_axis_height": 5.5,
"x_tick_frequency": 1,
"y_tick_frequency": 1,
"default_vector_field_config": {
"delta_x": .5,
"delta_y": .5,
"min_magnitude": 0,
"max_magnitude": 15,
"colors": [BLUE],
"length_func": lambda norm : norm/35,
"opacity": 1.0,
"vector_config": {
"stroke_width":2
},
},
"default_graph_style": {
"stroke_width": 2,
"stroke_color": WHITE,
},
}
def construct(self):
X = RIGHT*self.x_axis_width/(self.x_max- self.x_min)
Y = UP*self.y_axis_height/(self.y_max- self.y_min)
self.X=X ;self.Y=Y
self.setup_axes(animate=False)
fn_text=TexMobject(
r"\vec F = x^2\hat i-xy\hat j",
stroke_width=2.5
).set_color_by_gradient(
*self.default_vector_field_config["colors"]
)
fn_text.to_edge(TOP,buff=.1).shift(2*LEFT)
origin=self.graph_origin
v_field=self.get_vector_field(
lambda v: np.array([
(v[0]-origin[0])**2,
-(v[0]-origin[0])*(v[1]-origin[1]),
0,
]),
x_min= -.001+origin[0],
x_max= 5.4+origin[0],
y_min= -0+origin[1],
y_max= 5.5+origin[1],
)
self.add(v_field, fn_text)
self.play(Write(fn_text))
self.wait(2)
self.get_line_of_int()
self.get_dot_product_values()
self.wait(2)
self.remove(v_field,fn_text)
self.write_area_as_intgral_value()
self.wait(2)
def get_vector_field(self,func,**kwargs):
config = dict()
config.update(self.default_vector_field_config)
config.update(kwargs)
vector_field= VectorField(func,**config)
self.vector_field=vector_field
return vector_field
def get_line_of_int(self):
line_of_int_text=TextMobject(
r"Line of integration is\\",
"$\\vec r(t)=\cos(t)\hat i+\sin(t)\hat j$"
)
line_of_int_text[1].set_color(PINK)
line_of_int_text.to_corner(UR,buff=.8)
line_of_int= self.get_graph(
lambda x : np.sqrt(1-x**2),
x_min=1,
x_max=0,
)
line_of_int.set_style(
stroke_width=3,
stroke_color=PINK,
)
self.play(Write(line_of_int_text))
self.wait(.5)
self.play(ShowCreation(line_of_int),run_time=2)
# self.add(line_of_int)
self.line_of_int=line_of_int
self.line_of_int_text=line_of_int_text
def get_dot_product_values(self):
t_tracker = ValueTracker(0)
self.t_tracker = t_tracker
self.get_vector_and_tangent()
self.get_dot_product_graph()
self.wait(1.5)
self.play(ApplyMethod(
self.t_tracker.set_value, PI/6,
rate_func=linear,
run_time=2.5,
)
)
self.wait(1)
self.play(ApplyMethod(
self.t_tracker.set_value, PI/2,
rate_func=linear,
run_time=4,
)
)
self.dot_prod_graph.suspend_updating()
def get_vector_and_tangent(self):
vect_tangent_text=TextMobject(
"Get the"," vector",r" and\\"," tangent",
" on the"," line"
)
vect_tangent_text.set_color_by_tex_to_color_map({
"tangent": ORANGE, "vector": YELLOW, "line":PINK
})
vect_tangent_text.to_corner(UR,buff=.8)
self.vect_tangent_text= vect_tangent_text
self.play(FadeOut(self.axes))
self.remove(self.line_of_int_text)
self.play(Write(vect_tangent_text))
self.show_vector()
self.show_tangent()
self.wait(1.3)
def show_vector(self):
t = self.t_tracker.get_value
vect_label=TextMobject(
"$\\vec F(x_i,y_i)$",
color=YELLOW,
stroke_width=2
).scale(.8)
vector = always_redraw( lambda:
self.vector_field.get_vector(
self.coords_to_point(
np.cos(t()), np.sin(t())
),
stroke_width=6,
max_stroke_width_to_length_ratio= 8,
).set_color(YELLOW),
)
vect_label.next_to(vector,RIGHT,buff=.1)
vector_group= VGroup(vector,vect_label)
# self.add(vector_group)
self.play(Write(vector_group),run_time=1)
self.wait(.4)
self.vect_label = vect_label
self.vector_group= vector_group
def show_tangent(self):
tangent_label=TextMobject(
"$\\vec T(x_i,y_i)$",
color=ORANGE,
stroke_width=2
).scale(.8)
t = self.t_tracker.get_value
tangent = always_redraw(lambda:
Vector(
color=ORANGE,
stroke_width=6,
).scale(1).next_to(
self.coords_to_point(
np.cos(t()), np.sin(t())
),DR,buff=-.1
).rotate(
self.angle_of_tangent(
np.cos(t()),
self.line_of_int,
dx=-0.00001
),
about_point=self.coords_to_point(
np.cos(t()), np.sin(t())
)
)
)
tangent_label.next_to(tangent,UP,buff=.1)
tangent_group=VGroup(tangent,tangent_label)
# self.add(tangent_group)
self.play(Write(tangent_group))
self.wait(.6)
self.tangent_label=tangent_label
self.tangent_group=tangent_group
def get_dot_product_graph(self):
t = self.t_tracker.get_value
self.start_x= 1.3 ; self.end_x=2.3
t_axis= self.get_graph(
lambda x : 2.0/5,
x_min= self.start_x,
x_max= self.end_x,
).set_style(
stroke_width=4,
)
dot_prod_axis= Vector(3*UP).next_to(
t_axis,LEFT,buff=-.1
).set_color(GREEN)
dot_prod_label=TexMobject(
"\\vec F","\\cdot","\\vec T",
stroke_width= 1.5,
).next_to(dot_prod_axis,UP).scale(.8)
dot_prod_label[0].set_color(YELLOW)
dot_prod_label[2].set_color(ORANGE)
dot_prod_graph_axes= VGroup(t_axis,dot_prod_axis)
self.write_about_graph()
self.wait(1)
# self.add(dot_prod_graph_axes)
self.play(Write(dot_prod_graph_axes))
self.show_the_parameter(t,t_axis)
self.wait(.6)
self.play(ReplacementTransform(
self.vect_label,dot_prod_label[0]
))
self.play(ReplacementTransform(
self.tangent_label,dot_prod_label[1:3]
))
self.show_graph_area(t_axis)
self.dot_prod_graph_axes= dot_prod_graph_axes
self.dot_prod_label= dot_prod_label
def write_about_graph(self):
graph_text=TextMobject(
"Graph",r" of the "," vector",r" $-$\\ ",
r"tangent",r" dot product\\",
" with the parameter ","$t$"
)
graph_text.set_color_by_tex_to_color_map({
"Graph":GREEN, "vector": YELLOW,
"tangent":ORANGE, "$t$":RED
})
graph_text.to_corner(UR,buff=.5)
self.graph_text=graph_text
self.remove(self.vect_tangent_text)
self.play(Write(graph_text),run_time=4)
def show_the_parameter(self,t,t_axis):
t_dot=Dot(color=RED).next_to(t_axis,LEFT,buff=0)
t_dot.add_updater(lambda obj :
obj.move_to(self.c2g([t(),0])
))
t_text=TextMobject("$t$=").next_to(t_dot,UP,buff=.25)
t_val=always_redraw(
lambda: DecimalNumber(
t()/PI,
color=GOLD
).next_to(t_text,RIGHT,buff=0).scale(.8)
)
t_label=VGroup(
t_text,t_val
).set_color(RED)
pi = TexMobject(
"\\pi ",
color=GOLD,
).next_to(t_val,RIGHT,buff=0.05)
t_label.add(pi)
t_label.add_updater(lambda label :
label.next_to(t_dot,UP)
)
t_group=VGroup(t_dot,t_label)
# self.add(t_group)
self.play(Write(t_group))
self.t_group= t_group
def show_graph_area(self,t_axis):
t = self.t_tracker.get_value
dot_prod_graph= always_redraw(lambda: Polygon(
*[
self.c2g([t,-2*np.cos(t)**2*np.sin(t)])
for t in np.arange(0,t(),0.01)
],
*[
self.c2g([t,0])
for t in [ t(),0 ]
],
stroke_width=2.5,
fill_color=TEAL_D,
fill_opacity=.6,
))
self.add(dot_prod_graph)
self.dot_prod_graph=dot_prod_graph
def c2g(self,coord):
""" get points for the dot product graph
from its coordinates"""
return self.coords_to_point(
self.start_x+coord[0]/(PI/2),
2.0/5+coord[1]/2,
)
def write_area_as_intgral_value(self):
area_text=TextMobject(
"Value of the "," line"," integral in the",
r"Vector field\\",
"is equal to this ","area"
)
area_text.set_color_by_tex_to_color_map({
"Vector field": BLUE, "line":PINK, "area":TEAL_C
})
area_text.to_edge(TOP,buff=MED_SMALL_BUFF)
self.play(FadeOut(VGroup(
self.line_of_int,
self.vector_group,
self.tangent_group,
self.t_group,
self.dot_prod_graph_axes,
self.dot_prod_label,
self.graph_text
)
))
area= self.dot_prod_graph.copy().scale(1.3)
area.next_to(area_text,DOWN,buff=1.5)
# self.add(area_text)
self.play(Write(area_text),run_time=4)
self.play(ReplacementTransform(
self.dot_prod_graph,
area
))
self.wait(.5)
#uploaded by Somnath Pandit.FSF2020_Line_Integrals
|
