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Diffstat (limited to 'FSF-2020/calculus-of-several-variables/div-curl-grad-and-all-that/Greens Theorem/GreensTheorem_file1_ftc-analogue.py')
| -rw-r--r-- | FSF-2020/calculus-of-several-variables/div-curl-grad-and-all-that/Greens Theorem/GreensTheorem_file1_ftc-analogue.py | 265 |
1 files changed, 265 insertions, 0 deletions
diff --git a/FSF-2020/calculus-of-several-variables/div-curl-grad-and-all-that/Greens Theorem/GreensTheorem_file1_ftc-analogue.py b/FSF-2020/calculus-of-several-variables/div-curl-grad-and-all-that/Greens Theorem/GreensTheorem_file1_ftc-analogue.py new file mode 100644 index 0000000..222663b --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/div-curl-grad-and-all-that/Greens Theorem/GreensTheorem_file1_ftc-analogue.py @@ -0,0 +1,265 @@ +from manimlib.imports import * + + +def vector_field_func(coordinate): + x,y = coordinate[:2] + return np.array([ + x, + y, + 0 + ]) +def curl(coordinate): + x,y = coordinate[:2] + U = (x**2 + y**2) + return np.array([ + -y/(x**2 + y**2), + x/(x**2 + y**2), + 0 + ]) + + + + + +class GreensVisual(Scene): + + def construct(self): + axes_config = {"x_min": -6, + "x_max": 6, + "y_min": -6, + "y_max": 6, + "z_axis_config": {}, + "z_min": -1, + "z_max": 1, + "z_normal": DOWN, + "light_source": 9 * DOWN + 7 * LEFT + 10 * OUT, + "number_line_config": { + "include_tip": False, + }, + } + + axes = Axes(**axes_config) + + field = VectorField(vector_field_func).fade(0.5) + self.add(field) + + title = TexMobject(r"\textit{According to Green's Theorem, }").shift(3*UP) + + eq1 = TexMobject(r"\int_{C} \vec F . dr = \int \int_{D} \nabla \times \vec F dA").shift(3*DOWN) + eq5 = TexMobject(r"\int_{C} \vec F . dr = \int \int_{D} \nabla \times \vec F dA").shift(3*DOWN) + + generalisation = TexMobject() + + eq2 = TexMobject(r"\int_{C} \vec F . dr = \int_{C_1} \vec F . dr + \int_{C_{2}} \vec F . dr").shift(3*DOWN) + eq3 = TexMobject(r"\int_{C} \vec F . dr = \int_{C_{1}} \vec F . dr + \int_{C_{2}} \vec F . dr + \int_{C_{3}} \vec F . dr + \int_{C_{4}} \vec F . dr...").shift(3*DOWN) + eq4 = TexMobject(r"\int_{C_{r}} \vec F dr \approx \int\int_{D} \nabla \times \vec F dA").shift(3*DOWN) + eq = TexMobject(r"\int_{C_{r}} \textit{macroscopic curl} = \int\int_{D} \text{sum of all microscopic curls}").shift(3*UP) + + text_1 = TexMobject(r"\textit{Split C into 2 parts and calculate curl of each one of the smaller regions seperately}").shift(3*UP) + #text_2 = TexMobject(r"\textit{}").shift(3*UP) + text_3 = TexMobject(r"\textit{By splitting C into n segments, the area of each region approaches the limit 0}", r"\textit{The macroscopic circulation along the curve }", r"\textit{is equivalent to the sum of microscopic circulation of all these small regions }") + text_3[0].move_to(3.8*UP) + text_3[1].set_color(YELLOW_E).next_to(text_3[0], DOWN, buff = SMALL_BUFF) + text_3[2].set_color(BLUE_E).next_to(text_3[1], DOWN, buff = SMALL_BUFF) + + + + curl_rep_1 = StreamLines( + curl, + virtual_time=4, + min_magnitude=0, + max_magnitude=2, + dt = 0.1, + x_min = -0.5, x_max = 0.5, y_min = -0.5, y_max = 0.5, + ).set_color_by_gradient([BLUE_E, TEAL, WHITE]) + flow_1 = AnimatedStreamLines( + curl_rep_1, + line_anim_class=ShowPassingFlashWithThinningStrokeWidth + ) + + static = VMobject() + for p in range(0, 8, 4): + curl_rep_n = [*StreamLines( + curl, + virtual_time=2, + min_magnitude=0, + max_magnitude=1, + dt = 0.1, + x_min = -0.5, x_max = 0.5, y_min = -0.5, y_max = 0.5, + ).scale(0.5).move_to(np.array([-2+p, 0,0]))] + static.add(*curl_rep_n) + static_1 = VMobject() + for p in range(-3, 4, 2): + curl_rep_1 = [*StreamLines( + curl, + virtual_time=2, + min_magnitude=0, + max_magnitude=1, + dt = 0.1, + x_min = -0.5, x_max = 0.5, y_min = -0.5, y_max = 0.5, + ).scale(0.25).move_to(np.array([p, 0.6,0]))] + static_1.add(*curl_rep_1) + + static_2 = VMobject() + for p in range(-3, 4, 2): + curl_rep_2 = [*StreamLines( + curl, + virtual_time=2, + min_magnitude=0, + max_magnitude=1, + dt = 0.1, + x_min = -0.5, x_max = 0.5, y_min = -0.5, y_max = 0.5, + ).scale(0.25).move_to(np.array([p, -0.6,0]))] + static_2.add(*curl_rep_2) + + + + surface_6 = ParametricSurface( + self.surface, + u_min=-3, + u_max=3, + v_min=-3, + v_max=3, + fill_color=BLACK, + checkerboard_colors=[BLACK, BLACK], + stroke_color=BLUE_E, + resolution = [64,64] + ).set_fill(opacity=0.2).scale(1.5) + + boundary = ParametricSurface( + self.surface, + u_min=-3, + u_max=3, + v_min=-3, + v_max=3, + fill_color=BLACK, + checkerboard_colors=[BLACK, BLACK], + stroke_color=YELLOW_E, + resolution = [2,1] + ).set_fill(opacity=0).scale(1.75) + + + + + g1 = VGroup(surface_1, c) + g2 = VGroup( c1, c2, text_1) + g3 = VGroup(c1a, c2a, c3a, c4a) + + tr = Ellipse(width = 9, height = 3) + line = Line(tr.get_center()+1.5*UP, tr.get_center()+1.5*DOWN) + b = VMobject(stroke_color = "#F4EDED") + b.set_points_smoothly([tr.get_center()+1.5*UP, np.array([-2.25, 1.26, 0]), tr.get_center()+4.5*LEFT, np.array([-2.25, -1.26, 0]), tr.get_center()+1.5*DOWN]) + + + self.add(title) + self.play(ShowCreation(g1), ShowCreation(eq1)) + self.wait(3) + self.remove(flow_1) + self.play(ShowCreation(surface_2), ReplacementTransform(eq1, eq2)) + self.remove(g1) + self.wait() + self.play(ReplacementTransform(surface_2, surface_3), ReplacementTransform(eq2, eq3)) + self.wait() + self.wait() + self.play(FadeOut(surface_3), ShowCreation(surface_4), ReplacementTransform(eq3, eq4), ReplacementTransform(title, eq)) + self.play(FadeOut(surface_4), ShowCreation(surface_5)) + self.play(FadeOut(surface_5), ShowCreation(surface_6)) + self.wait() + #self.add(tr, line) + self.wait() + grd = ScreenGrid() + + g = ParametricFunction(func, t_min = 0, t_max = 2*PI).scale(1.5) + self.add(grd, g) + self.wait() + + +def circ(coordinate): + x,y = coordinate[:2] + for x in range(0, -5) and y in range(-1,1): + cr = Ellipse() + return cr + +def func(t): + return np.array([ + np.sin(t), + np.cos(t), + 0]) + +def surf(t,u): + return np.array([ + u*np.sin(t), + np.cos(t), + 0]) + +class Analogy(GraphScene): + CONFIG = { + "x_min": -1, + "x_max": 4, + "y_min": 0, + "y_max": 2, + "y_tick_frequency": 2.5, + "n_rect_iterations": 6, + "default_right_x": 3, + } + + def construct(self): + + + ftc = TexMobject(r"\int_a^b f'(x) \ dx", r" = f(b) - f(a)").shift(3*UP).set_color("#F9DB6D").scale(0.7) + greens = TexMobject(r"\int \int_{R} curl \left(\vec F \right) \ dxdy", r" = \int_{C} \vec F \ dr").shift(3*UP).set_color("#F9DB6D").scale(0.7) + ftc[0].set_color("#36827F") + greens[1].set_color("#36827F") + + + two_to_one = TexMobject(r"\textit{2D region} \to", r"\textit{1D curve}").shift(3.6*DOWN).scale(0.7).set_color("#F9DB6D") + one_to_zero = TexMobject(r"\textit{1D curve}", r" \to \textit{0D points}").shift(3.6*DOWN).set_color("#F9DB6D").scale(0.7) + two_to_one[1].set_color("#36827F") + one_to_zero[0].set_color("#36827F") + greens_title = TexMobject(r"\textit{Green's Theorem}").scale(0.8).next_to(two_to_one, UP, buff = SMALL_BUFF).set_color("#F4EDED") + ftc_title = TexMobject(r"\textit{Fundamental Theorem of Calculus}").scale(0.8).next_to(two_to_one, UP, buff = SMALL_BUFF).set_color("#F4EDED") + + surf= VMobject(fill_color = "#ED6A5A", stroke_color = "#ED6A5A", fill_opacity = 0.6) + surf.set_points_smoothly([np.array([-2, 1.8,0]),np.array([-1.6, 0.5,0]),np.array([-3.2, -1.2,0]),np.array([2.6, -1.5,0]),np.array([1, 0,0]),np.array([3.5,2.3, 0]), np.array([-2,1.8, 0])]) + dot = Dot(np.array([-2,1.8, 0])).set_color("#F4EDED") + boundary = VMobject(stroke_color = "#F4EDED") + boundary.set_points_smoothly([np.array([-2, 1.8,0]),np.array([-1.6, 0.5,0]),np.array([-3.2, -1.2,0]),np.array([2.6, -1.5,0]),np.array([1, 0,0]),np.array([3.5,2.3, 0]), np.array([-2,1.8, 0])]) + c = TexMobject(r"C").next_to(surf,RIGHT+UP).set_color("#F4EDED") + r = TexMobject(r"R").move_to(np.array([-0.2, 0.6, 0])).set_color("#F4EDED") + + self.play(ShowCreation(surf), ShowCreation(r)) + self.wait(2) + self.play(ShowCreation(boundary), MoveAlongPath(dot, boundary), Write(c), Write(greens),Write(greens_title), run_time= 1.5) + self.wait(2) + self.play(ReplacementTransform(surf, boundary), FadeOut(r), Write(two_to_one), FadeOut(dot)) + self.wait(2) + + self.setup_axes() + + grapher = self.get_graph(self.funk) + grapher.set_color("#E94F37") + l1 = self.get_vertical_line_to_graph(1, grapher, color = "#F4EDED") + l2 =self.get_vertical_line_to_graph(3, grapher, color = "#F4EDED") + label_coord_1 = self.input_to_graph_point(1,grapher) + label_coord_2 = self.input_to_graph_point(3,grapher) + + + a = TexMobject(r"a").next_to(label_coord_1,RIGHT+UP).set_color("#F4EDED") + b = TexMobject(r"b").next_to(label_coord_2,RIGHT+UP).set_color("#F4EDED") + + + + + + point_a = Dot(label_coord_1).set_color("#827081") + point_b = Dot(label_coord_2).set_color("#827081") + + + self.play(ReplacementTransform(boundary, grapher), FadeOut(c), FadeIn(a), FadeIn(b), FadeIn(point_a), FadeIn(point_b), ReplacementTransform(greens, ftc), ReplacementTransform(greens_title, ftc_title)) + self.wait(2) + self.play(Uncreate(grapher), ReplacementTransform(two_to_one, one_to_zero)) + self.wait(2) + + + def funk(self,x): + return 0.2*(x-2)**2 +1
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