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Simple gradient worked example

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This commit is contained in:
Grant Sanderson
2018-08-06 10:45:48 -07:00
parent 1df53ef6e9
commit 4e5720d0ab
1 changed files with 184 additions and 21 deletions
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205
active_projects/gradient.py
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@ -155,7 +155,11 @@ class ShowGradient(Scene):
self.wait()
class SliceGraphTwoWays(ExternallyAnimatedScene):
class ExampleGraphHoldXConstant(ExternallyAnimatedScene):
pass
class ExampleGraphHoldYConstant(ExternallyAnimatedScene):
pass
@ -166,26 +170,27 @@ class TakePartialDerivatives(Scene):
"y": RED,
}
func_tex = TexMobject(
"f(x, y)", "=", "e^{", "-x^2 + \\cos(2y)}",
"f", "(", "x", ",", "y", ")", "=",
"e^{", "-x^2", "+ \\cos(2y)}",
tex_to_color_map=tex_to_color_map
)
partial_x = TexMobject(
"{\\partial f \\over \\partial x}", "=",
"\\left( e^{-x^2 + \\cos(2y)} \\right)",
"(-2x)",
tex_to_color_map=tex_to_color_map
"{\\partial", "f", "\\over", "\\partial", "x}", "=",
"\\left(", "e^", "{-x^2", "+ \\cos(2y)}", "\\right)",
"(", "-2", "x", ")",
tex_to_color_map=tex_to_color_map,
)
partial_y = TexMobject(
"{\\partial f \\over \\partial y}", "=",
"\\left( e^{-x^2 + \\cos(2y)} \\right)",
"(-\\sin(2y) \\cdot 2)",
tex_to_color_map=tex_to_color_map
"{\\partial", "f", "\\over", "\\partial", "y}", "=",
"\\left(", "e^", "{-x^2", "+ \\cos(", "2", "y)}", "\\right)",
"(", "-\\sin(", "2", "y)", "\\cdot 2", ")",
tex_to_color_map=tex_to_color_map,
)
partials = VGroup(partial_x, partial_y)
for mob in func_tex, partials:
mob.scale(1.5)
func_tex.move_to(2 * UP + LEFT)
func_tex.move_to(2 * UP + 3 * LEFT)
for partial in partials:
partial.next_to(func_tex, DOWN, buff=LARGE_BUFF)
top_eq_x = func_tex.get_part_by_tex("=").get_center()[0]
@ -208,23 +213,181 @@ class TakePartialDerivatives(Scene):
Vector(0.5 * DOWN).next_to(rect, UP, SMALL_BUFF)
for rect in terms.rects
])
treat_as_constant
self.add(func_tex, partial_x)
self.add(exp_rect)
self.add(xs.rects)
self.add(ys.rects)
self.add(xs.arrows)
self.add(ys.arrows)
treat_as_constant = TextMobject("Treat as a constant")
treat_as_constant.next_to(ys.arrows[1], UP)
# Start to show partial_x
self.play(FadeInFromDown(func_tex))
self.wait()
self.play(
ReplacementTransform(func_tex[0].copy(), partial_x[1]),
Write(partial_x[0]),
Write(partial_x[2:4]),
Write(partial_x[6]),
)
self.play(
ReplacementTransform(func_tex[2].copy(), partial_x[4])
)
self.wait()
# Label y as constant
self.play(LaggedStart(ShowCreation, ys.rects))
self.play(
LaggedStart(GrowArrow, ys.arrows, lag_ratio=0.8),
Write(treat_as_constant)
)
self.wait(2)
# Perform partial_x derivative
self.play(FadeIn(exp_rect), Animation(func_tex))
self.wait()
pxi1 = 8
pxi2 = 15
self.play(
ReplacementTransform(
func_tex[7:].copy(),
partial_x[pxi1:pxi2],
),
FadeIn(partial_x[pxi1 - 1:pxi1]),
FadeIn(partial_x[pxi2]),
)
self.wait(2)
self.play(
ReplacementTransform(
partial_x[10:12].copy(),
partial_x[pxi2 + 2:pxi2 + 4],
path_arc=-(TAU / 4)
),
FadeIn(partial_x[pxi2 + 1]),
FadeIn(partial_x[-1]),
)
self.wait(2)
# Swap out partial_x for partial_y
self.play(
FadeOutAndShiftDown(partial_x),
FadeOut(ys.rects),
FadeOut(ys.arrows),
FadeOut(treat_as_constant),
FadeOut(exp_rect),
Animation(func_tex)
)
self.play(FadeInFromDown(partial_y[:7]))
self.wait()
treat_as_constant.next_to(xs.arrows[1], UP, SMALL_BUFF)
self.play(
LaggedStart(ShowCreation, xs.rects),
LaggedStart(GrowArrow, xs.arrows),
Write(treat_as_constant),
lag_ratio=0.8
)
self.wait()
# Show same outer derivative
# Finish y derivative
self.play(
ReplacementTransform(
func_tex[7:].copy(),
partial_x[pxi1:pxi2],
),
FadeIn(partial_x[pxi1 - 2:pxi1]),
FadeIn(partial_x[pxi2]),
)
self.wait()
self.play(
ReplacementTransform(
partial_y[12:16].copy(),
partial_y[pxi2 + 3:pxi2 + 7],
path_arc=-(TAU / 4)
),
FadeIn(partial_y[pxi2 + 2]),
FadeIn(partial_y[-1]),
)
self.wait()
self.play(ReplacementTransform(
partial_y[-5].copy(),
partial_y[-2],
path_arc=-PI
))
self.wait()
class ShowDerivativeAtExamplePoint(Scene):
def construct(self):
pass
tex_to_color_map = {
"x": BLUE,
"y": RED,
}
func_tex = TexMobject(
"f", "(", "x", ",", "y", ")", "=",
"e^{", "-x^2", "+ \\cos(2y)}",
tex_to_color_map=tex_to_color_map
)
gradient_tex = TexMobject(
"\\nabla", "f", "(", "x", ",", "y", ")", "=",
tex_to_color_map=tex_to_color_map
)
partial_vect = Matrix([
["{\\partial f / \\partial x}"],
["{\\partial f / \\partial y}"],
])
partial_vect.get_mob_matrix()[1, 0][-1].set_color(BLUE)
partial_vect.get_mob_matrix()[0, 0][-1].set_color(RED)
result_vector = self.get_result_vector("x", "y")
gradient = VGroup(
gradient_tex,
partial_vect,
TexMobject("="),
result_vector
)
gradient.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
func_tex.to_edge(UP)
gradient.next_to(func_tex, DOWN, buff=LARGE_BUFF)
example_lhs = TexMobject(
"\\nabla", "f", "(", "1", ",", "3", ")", "=",
tex_to_color_map={"1": BLUE, "3": RED},
)
example_result_vector = self.get_result_vector("1", "3")
example_rhs = DecimalMatrix([[-1.92], [0.54]])
example = VGroup(
example_lhs,
example_result_vector,
TexMobject("="),
example_rhs,
)
example.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
example.next_to(gradient, DOWN, LARGE_BUFF)
self.add(func_tex, gradient)
self.wait()
self.play(
ReplacementTransform(gradient_tex.copy(), example_lhs),
ReplacementTransform(result_vector.copy(), example_result_vector),
)
self.wait()
self.play(Write(example[2:]))
self.wait()
def get_result_vector(self, x, y):
result_vector = Matrix([
["e^{-%s^2 + \\cos(2\\cdot %s)} (-2\\cdot %s)" % (x, y, x)],
["e^{-%s^2 + \\cos(2\\cdot %s)} \\big(-\\sin(2\\cdot %s) \\cdot 2\\big)" % (x, y, y)],
], v_buff=1.2, element_alignment_corner=ORIGIN)
x_terms = VGroup(
result_vector.get_mob_matrix()[0, 0][2],
result_vector.get_mob_matrix()[1, 0][2],
result_vector.get_mob_matrix()[0, 0][-2],
)
y_terms = VGroup(
result_vector.get_mob_matrix()[0, 0][11],
result_vector.get_mob_matrix()[1, 0][11],
result_vector.get_mob_matrix()[1, 0][-5],
)
x_terms.set_color(BLUE)
y_terms.set_color(RED)
return result_vector