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DivCurlDotCross
This commit is contained in:
Grant Sanderson
parent
719f24c0dc
commit
3bfc515411
1 changed files with 306 additions and 3 deletions
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@ -1290,7 +1290,7 @@ class DefineDivergence(ChangingElectricField):
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"delta_x": 0.125,
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"delta_y": 0.125,
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},
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"virtual_time": 2,
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"virtual_time": 5,
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"n_anchors_per_line": 10,
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"min_magnitude": 0,
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"max_magnitude": 1,
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@ -2647,7 +2647,7 @@ class ShowTwoPopulations(Scene):
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return count
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class PhaseSpaceOfPopulationModel(ShowTwoPopulations, PiCreatureScene):
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class PhaseSpaceOfPopulationModel(ShowTwoPopulations, PiCreatureScene, MovingCameraScene):
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CONFIG = {
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"origin": 5 * LEFT + 2.5 * DOWN,
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"vector_field_config": {
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@ -2660,6 +2660,10 @@ class PhaseSpaceOfPopulationModel(ShowTwoPopulations, PiCreatureScene):
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"flow_time": 10,
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}
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def setup(self):
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MovingCameraScene.setup(self)
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PiCreatureScene.setup(self)
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def construct(self):
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self.add_axes()
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self.add_example_point()
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@ -2749,8 +2753,8 @@ class PhaseSpaceOfPopulationModel(ShowTwoPopulations, PiCreatureScene):
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self.add(*pop_sizes_updates)
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self.play(
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dot.restore,
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VFadeIn(coord_pair),
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UpdateFromAlphaFunc(pop_sizes, lambda m, a: m.set_fill(opacity=a)),
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VFadeIn(coord_pair)
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)
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self.wait()
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self.play(Write(phase_space))
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@ -2849,6 +2853,7 @@ class PhaseSpaceOfPopulationModel(ShowTwoPopulations, PiCreatureScene):
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self.play(
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Animation(dot),
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vector_field.set_fill, {"opacity": 0.2},
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Animation(self.differential_equation_group),
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GrowArrow(dot_vector),
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randy.change, "pondering",
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)
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@ -2924,3 +2929,301 @@ class PhaseSpaceOfPopulationModel(ShowTwoPopulations, PiCreatureScene):
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self.add(stream_line_animation)
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self.add_foreground_mobjects(vector_field)
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self.wait(self.flow_time)
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self.play(
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self.camera_frame.scale, 1.5, {"about_point": self.origin},
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run_time=self.flow_time,
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)
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self.wait(self.flow_time)
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class AskAboutComputation(TeacherStudentsScene):
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def construct(self):
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self.student_says(
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"Sure, but how do you \\\\" +
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"\\emph{compute} $\\textbf{div}$ and $\\textbf{curl}$?",
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target_mode="sassy",
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)
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self.change_student_modes(
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"confused", "sassy", "angry",
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added_anims=[self.teacher.change, "guilty"]
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)
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self.wait()
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self.teacher_says(
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"Are you familiar \\\\" +
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"with my work \\\\" +
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"at Khan Academy?",
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target_mode="speaking",
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bubble_kwargs={"width": 4, "height": 3}
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)
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self.change_student_modes(
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* 3 * ["pondering"],
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look_at_arg=self.screen
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)
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self.wait(5)
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class NablaNotation(PiCreatureScene, MovingCameraScene):
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CONFIG = {
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"default_pi_creature_kwargs": {
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"color": GREY_BROWN,
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},
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"default_pi_creature_start_corner": DL,
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}
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def setup(self):
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MovingCameraScene.setup(self)
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PiCreatureScene.setup(self)
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def construct(self):
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self.show_notation()
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self.show_expansion()
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self.zoom_out()
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def show_notation(self):
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morty = self.pi_creature
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tex_to_color_map = {
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"\\text{div}": BLUE,
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"\\nabla \\cdot": BLUE,
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"\\text{curl}": YELLOW,
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"\\nabla \\times": YELLOW,
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}
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div_equation = TexMobject(
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"\\text{div} \\, \\textbf{F} = \\nabla \\cdot \\textbf{F}",
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tex_to_color_map=tex_to_color_map
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)
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div_nabla = div_equation.get_part_by_tex("\\nabla")
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curl_equation = TexMobject(
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"\\text{curl} \\, \\textbf{F} = \\nabla \\times \\textbf{F}",
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tex_to_color_map=tex_to_color_map
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)
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curl_nabla = curl_equation.get_part_by_tex("\\nabla")
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equations = VGroup(div_equation, curl_equation)
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equations.arrange_submobjects(DOWN, buff=LARGE_BUFF)
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equations.next_to(morty, UP, 2)
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equations.to_edge(LEFT)
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self.play(
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FadeInFromDown(div_equation),
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morty.change, "raise_right_hand"
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)
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self.wait()
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self.play(WiggleOutThenIn(div_nabla, scale_value=1.5))
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self.wait()
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self.play(
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FadeInFromDown(curl_equation),
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morty.change, "raise_left_hand"
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)
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self.wait()
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self.play(WiggleOutThenIn(curl_nabla, scale_value=1.5))
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self.wait()
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self.equations = equations
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def show_expansion(self):
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equations = self.equations
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morty = self.pi_creature
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nabla_vector = Matrix([
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["\\partial \\over \\partial x"],
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["\\partial \\over \\partial y"],
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], v_buff=1.5)
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F_vector = Matrix([
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["\\textbf{F}_x"],
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["\\textbf{F}_y"],
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], v_buff=1.2)
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nabla_vector.match_height(F_vector)
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div_lhs, curl_lhs = lhs_groups = VGroup(*[
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VGroup(
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nabla_vector.deepcopy(),
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TexMobject(tex).scale(1.5),
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F_vector.copy(),
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TexMobject("=")
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)
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for tex in "\\cdot", "\\times"
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])
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colors = [BLUE, YELLOW]
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for lhs, color in zip(lhs_groups, colors):
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lhs.arrange_submobjects(RIGHT, buff=MED_SMALL_BUFF)
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VGroup(lhs[0].brackets, lhs[1]).set_color(color)
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div_lhs.to_edge(UP)
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curl_lhs.next_to(div_lhs, DOWN, buff=LARGE_BUFF)
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div_rhs = TexMobject(
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"{\\partial F_x \\over \\partial x} + " +
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"{\\partial F_y \\over \\partial y}"
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)
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curl_rhs = TexMobject(
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"{\\partial F_y \\over \\partial x} - " +
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"{\\partial F_x \\over \\partial y}"
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)
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rhs_groups = VGroup(div_rhs, curl_rhs)
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for rhs, lhs in zip(rhs_groups, lhs_groups):
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rhs.next_to(lhs, RIGHT)
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for rhs, tex, color in zip(rhs_groups, ["div", "curl"], colors):
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rhs.rect = SurroundingRectangle(rhs, color=color)
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rhs.label = TexMobject(
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"\\text{%s}" % tex,
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"\\, \\textbf{F}"
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)
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rhs.label.set_color(color)
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rhs.label.next_to(rhs.rect, UP)
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for i in 0, 1:
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self.play(
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ReplacementTransform(
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equations[i][2].copy(),
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lhs_groups[i][0].brackets
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),
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ReplacementTransform(
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equations[i][3].copy(),
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lhs_groups[i][2],
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),
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morty.change, "pondering",
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*[
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GrowFromPoint(mob, equations[i].get_right())
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for mob in [
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lhs_groups[i][0].get_entries(),
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lhs_groups[i][1],
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lhs_groups[i][3]
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]
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],
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run_time=2
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)
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self.wait()
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self.wait()
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for rhs in rhs_groups:
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self.play(
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Write(rhs),
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morty.change, 'confused'
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)
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self.play(
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ShowCreation(rhs.rect),
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FadeInFromDown(rhs.label),
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)
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self.wait()
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self.play(morty.change, "erm")
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self.wait(3)
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def zoom_out(self):
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screen_rect = self.camera_frame.copy()
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screen_rect.set_stroke(WHITE, 3)
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screen_rect.scale(1.01)
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words = TextMobject("Something deeper at play...")
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words.scale(1.3)
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words.next_to(screen_rect, UP)
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self.add(screen_rect)
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self.play(
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self.camera_frame.scale_to_fit_height, FRAME_HEIGHT + 3,
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Write(words, rate_func=squish_rate_func(smooth, 0.3, 1)),
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run_time=2,
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)
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self.wait()
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class DivCurlDotCross(Scene):
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def construct(self):
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rects = VGroup(*[
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ScreenRectangle(height=2.5)
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for n in range(4)
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])
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rects.arrange_submobjects_in_grid(n_rows=2, buff=LARGE_BUFF)
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rects[2:].shift(MED_LARGE_BUFF * DOWN)
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titles = VGroup(*map(TextMobject, [
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"Divergence", "Curl",
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"Dot product", "Cross product"
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]))
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for title, rect in zip(titles, rects):
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title.next_to(rect, UP)
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self.add(rects, titles)
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class ShowDotProduct(MovingCameraScene):
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CONFIG = {
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"prod_tex": "\\cdot"
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}
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def construct(self):
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plane = NumberPlane()
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v1 = Vector(RIGHT, color=BLUE)
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v2 = Vector(UP, color=YELLOW)
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dot_product = TexMobject(
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"\\vec{\\textbf{v}}", self.prod_tex,
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"\\vec{\\textbf{w}}", "="
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)
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dot_product.set_color_by_tex_to_color_map({
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"textbf{v}": BLUE,
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"textbf{w}": YELLOW,
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})
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dot_product.add_background_rectangle()
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dot_product.next_to(2.25 * UP, RIGHT)
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dot_product_value = DecimalNumber(
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1.0,
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include_background_rectangle=True,
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)
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dot_product_value.next_to(dot_product)
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dot_product_value_update = ContinualChangingDecimal(
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dot_product_value,
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lambda a: self.get_product(v1, v2),
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include_background_rectangle=True,
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)
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self.camera_frame.scale_to_fit_height(4)
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self.camera_frame.move_to(DL, DL)
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self.add(plane)
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self.add(dot_product, dot_product_value_update)
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self.add_additional_continual_animations(v1, v2)
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self.add_foreground_mobjects(v1, v2)
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for n in range(5):
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self.play(
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Rotate(v1, 45 * DEGREES, about_point=ORIGIN),
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Rotate(v2, -45 * DEGREES, about_point=ORIGIN),
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run_time=3,
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rate_func=there_and_back
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)
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self.wait(0.5)
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def get_product(self, v1, v2):
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return np.dot(v1.get_vector(), v2.get_vector())
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def add_additional_continual_animations(self):
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pass
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class ShowCrossProduct(ShowDotProduct):
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CONFIG = {
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"prod_tex": "\\times"
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}
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def get_product(self, v1, v2):
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return np.linalg.norm(
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np.cross(v1.get_vector(), v2.get_vector())
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)
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def add_additional_continual_animations(self, v1, v2):
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square = Square(
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stroke_color=YELLOW,
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stroke_width=3,
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fill_color=YELLOW,
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fill_opacity=0.2,
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)
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self.add(ContinualUpdateFromFunc(
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square,
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lambda s: s.set_points_as_corners([
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ORIGIN,
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v1.get_end(),
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v1.get_end() + v2.get_end(),
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v2.get_end(),
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])
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))
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class NewSceneName(Scene):
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def construct(self):
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pass
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