From ac83cc7b152273a6f6bd823028e7dec75a6f9fd1 Mon Sep 17 00:00:00 2001 From: nishanpoojary Date: Wed, 1 Jul 2020 11:56:42 +0530 Subject: Added updated multivariable-limits-and-continuity folder --- ...d_Continuity_of_Multivariable_Function_Quiz.pdf | Bin 0 -> 101435 bytes .../file1_epsilon_delta_defn.py | 179 +++++++++++++++++++++ .../file2_limit_approach_point.py | 66 ++++++++ .../file3_limit_approach_point_3d.py | 152 +++++++++++++++++ .../file4_limit_different_point.py | 115 +++++++++++++ .../file5_continuity_func.py | 115 +++++++++++++ .../gifs/file1_epsilon_delta_defn.gif | Bin 0 -> 1788321 bytes .../gifs/file2_limit_approach_point.gif | Bin 0 -> 47411 bytes .../gifs/file3_limit_approach_point_3d.gif | Bin 0 -> 2770965 bytes .../gifs/file4_limit_different_point.gif | Bin 0 -> 3044265 bytes .../gifs/file5_continuity_func.gif | Bin 0 -> 5035077 bytes 11 files changed, 627 insertions(+) create mode 100644 FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/Limits_and_Continuity_of_Multivariable_Function_Quiz.pdf create mode 100644 FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file1_epsilon_delta_defn.py create mode 100644 FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file2_limit_approach_point.py create mode 100644 FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file3_limit_approach_point_3d.py create mode 100644 FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file4_limit_different_point.py create mode 100644 FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file5_continuity_func.py create mode 100644 FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file1_epsilon_delta_defn.gif create mode 100644 FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file2_limit_approach_point.gif create mode 100644 FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file3_limit_approach_point_3d.gif create mode 100644 FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file4_limit_different_point.gif create mode 100644 FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file5_continuity_func.gif (limited to 'FSF-2020') diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/Limits_and_Continuity_of_Multivariable_Function_Quiz.pdf b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/Limits_and_Continuity_of_Multivariable_Function_Quiz.pdf new file mode 100644 index 0000000..99918e5 Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/Limits_and_Continuity_of_Multivariable_Function_Quiz.pdf differ diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file1_epsilon_delta_defn.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file1_epsilon_delta_defn.py new file mode 100644 index 0000000..803c122 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file1_epsilon_delta_defn.py @@ -0,0 +1,179 @@ +from manimlib.imports import * + +class EpsilonDelta(ThreeDScene): + def construct(self): + axes = ThreeDAxes() # creates a 3D Axis + + + sphere = ParametricSurface( + lambda u, v: np.array([ + 3*np.sin(u)*np.cos(v), + 3*np.sin(u)*np.sin(v), + 3*np.cos(u) + ]),u_min=0,u_max=PI/4,v_min=PI/2,v_max=PI,checkerboard_colors=[RED_D, RED_E], + resolution=(15, 32)).scale(1) + + + cylinder_z = ParametricSurface( + lambda u, v: np.array([ + 0.25*np.cos(TAU * v), + 1.8* (1 - u), + 0.25*np.sin(TAU * v) + + ]), + checkerboard_colors=[YELLOW_C, YELLOW_E], resolution=(6, 32)).fade(0.2).rotate(PI/4).move_to(np.array([-0.65,0.65,2.54])) + + + cylinder_x = ParametricSurface( + lambda u, v: np.array([ + 0.3*np.cos(TAU * v)-1, + 0.3*np.sin(TAU * v)+1, + 2.6*(1 - u) + ]), + checkerboard_colors=[BLUE_C, BLUE_E], resolution=(6, 32)).fade(0.2) + + + delta_circle = Circle(radius= 0.3, color = BLACK).shift(1*LEFT+1*UP).set_fill(GREEN_E, opacity = 0.5) + + epsilon_circle = [np.array([0.25*np.cos(i*DEGREES),0,0.25*np.sin(i*DEGREES)]) for i in range(361)] + + epsilon_circle_polygon = Polygon(*epsilon_circle, color = RED_E, fill_color = RED_E, fill_opacity = 0.5).rotate(PI/4).move_to(np.array([0,0,2.54])) + + + dot_circle = Dot().move_to(np.array([-1,1,0])).set_fill("#000080") + + dot_surface = Dot().rotate(-PI/4).scale(1.5).move_to(np.array([-1.2,1.2,2.7])).set_fill("#000080") + + dot_L_epsilon1 = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).rotate(PI/4).move_to(np.array([0,0,2.3])) + + dot_L_epsilon2 = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).rotate(PI/4).move_to(np.array([0,0,2.8])) + + dot_L = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#006400", fill_color = "#006400", fill_opacity = 1).rotate(PI/4).move_to(np.array([0,0,2.54])) + + + + self.add(axes) + + axis = TextMobject(r"X",r"Y",r"Z") + axis[0].move_to(6*RIGHT) + axis[1].move_to(6*UP) + axis[2].move_to(np.array([0,0,3.7])) + + self.add_fixed_orientation_mobjects(axis[2]) + self.add_fixed_orientation_mobjects(axis[0]) + self.add_fixed_orientation_mobjects(axis[1]) + + self.set_camera_orientation(phi=75*DEGREES,theta=135*DEGREES) + #self.set_camera_orientation(phi=80*DEGREES,theta=45*DEGREES) + + + self.play(ShowCreation(sphere),ShowCreation(delta_circle), ShowCreation(dot_circle)) + + temp_circle_center = TextMobject(r"$(a,b,0)$").scale(0.6).set_color(BLUE_C).move_to(1.7*LEFT+1.1*UP) + self.add_fixed_orientation_mobjects(temp_circle_center) + self.wait() + + delta_lab = TextMobject(r"$\delta$", r"$-$", "disk").scale(0.5).move_to(0.6*LEFT+1.7*UP) + delta_lab[0].set_color(PINK).scale(1.3) + delta_lab[1].set_color(ORANGE) + delta_lab[2].set_color(GREEN_E) + + self.add_fixed_orientation_mobjects(delta_lab) + + self.play(ShowCreation(dot_surface)) + + temp_curve_circle_center = TextMobject(r"$(a,b,L)$").scale(0.6).set_color("#006400").move_to(np.array([-2,1,2.7])) + self.add_fixed_orientation_mobjects(temp_curve_circle_center) + + + self.wait() + self.play(ShowCreation(cylinder_x), FadeOut(dot_surface)) + self.wait() + + self.move_camera(phi=0* DEGREES,theta=135*DEGREES) + self.wait() + + self.move_camera(phi=80* DEGREES,theta=225*DEGREES) + self.wait() + + self.play(FadeOut(delta_lab), ShowCreation(cylinder_z)) + self.wait() + + self.play(FadeOut(temp_circle_center), FadeOut(temp_curve_circle_center),ShowCreation(epsilon_circle_polygon)) + + self.move_camera(phi=80* DEGREES,theta=325*DEGREES) + + dot_L_epsilon1_lab = TextMobject(r"$L$", r"$-$", r"$\epsilon$").scale(0.6).move_to(np.array([-0.4,-0.4,2.3])) + dot_L_epsilon1_lab[0].set_color("#D4108A") + dot_L_epsilon1_lab[1].set_color("#006400") + dot_L_epsilon1_lab[2].set_color("#4DC8A1").scale(1.5) + + dot_L_epsilon2_lab = TextMobject(r"$L$", r"$+$", r"$\epsilon$").scale(0.6).move_to(np.array([-0.4,-0.4,2.8])) + dot_L_epsilon2_lab[0].set_color("#D4108A") + dot_L_epsilon2_lab[1].set_color("#006400") + dot_L_epsilon2_lab[2].set_color("#4DC8A1").scale(1.5) + + dot_L_lab = TextMobject(r"$L$").scale(0.6).set_color("#D4108A").move_to(np.array([-0.4,-0.4,2.54])) + + + self.play(ShowCreation(dot_L_epsilon1), ShowCreation(dot_L), ShowCreation(dot_L_epsilon2)) + self.add_fixed_orientation_mobjects(dot_L_epsilon1_lab, dot_L_epsilon2_lab, dot_L_lab) + self.wait(4) + + self.move_camera(phi=80* DEGREES,theta=45*DEGREES) + self.wait(2) + + + + + + + + + + + + + + + ''' + + + + + + + + + + + + delta_lab = TextMobject(r"$\delta - disk$") + delta_lab.scale(0.5) + delta_lab.set_color(PINK) + + self.play(ShowCreation(circle_center)) + self.add_fixed_in_frame_mobjects(temp_circle_center) + temp_circle_center.move_to(1.5*RIGHT) + self.play(Write(temp_circle_center)) + + self.play(ShowCreation(curve_circle_center)) + self.add_fixed_in_frame_mobjects(temp_curve_circle_center) + temp_curve_circle_center.move_to(1.9*UP+1*RIGHT) + self.play(Write(temp_curve_circle_center)) + + + self.add_fixed_in_frame_mobjects(delta_lab) + delta_lab.move_to(0.4*DOWN+1.7*RIGHT) + self.play(Write(delta_lab)) + + + + + + self.begin_ambient_camera_rotation(rate=0.2) + + self.play(ShowCreation(circle), ShowCreation(line1), ShowCreation(line2)) + self.play(ShowCreation(line3), ShowCreation(line4)) + self.wait(8) + ''' \ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file2_limit_approach_point.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file2_limit_approach_point.py new file mode 100644 index 0000000..57d1d45 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file2_limit_approach_point.py @@ -0,0 +1,66 @@ +from manimlib.imports import * + +class Limit(GraphScene): + CONFIG = { + "x_min": 0, + "x_max": 4, + "y_min": 0, + "y_max": 4, + "graph_origin": ORIGIN + 3* DOWN+4*LEFT, + "x_labeled_nums": list(range(0, 4)), + "y_labeled_nums": list(range(0, 5)), + } + def construct(self): + topic = TextMobject("Different paths of approach to limit point") + topic.scale(1.5) + topic.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE) + self.play(Write(topic)) + self.wait(1) + self.play(FadeOut(topic)) + + + + XTD = self.x_axis_width/(self.x_max- self.x_min) + YTD = self.y_axis_height/(self.y_max- self.y_min) + + self.setup_axes(animate = True) + + y_x = self.get_graph(lambda x : x, x_min = -1, x_max = 4) + y_x_lab = self.get_graph_label(y_x, label = r"y = x") + + y_xsquare = self.get_graph(lambda x : x*x, x_min = -1, x_max = 4) + y_xsquare_lab = self.get_graph_label(y_xsquare, label = r"y = x^2") + + y_1 = self.get_graph(lambda x : 1, x_min = -1, x_max = 4) + y_1_lab = self.get_graph_label(y_1, label = r"y = 1") + + y_2minusx = self.get_graph(lambda x : 2 - x, x_min = -1, x_max = 4, color = RED) + y_2minusx_lab = self.get_graph_label(y_2minusx, label = r"y = 2 - x") + + limit_point = Dot().shift(self.graph_origin+1*XTD*RIGHT+1*YTD*UP) + limit_point_lab = TextMobject(r"(1,1)") + limit_point_lab.next_to(limit_point, DOWN) + + self.play(ShowCreation(limit_point)) + self.play(Write(limit_point_lab)) + self.wait(1) + + self.play(ShowCreation(y_x)) + self.play(Write(y_x_lab)) + self.wait(1) + + self.play(ShowCreation(y_xsquare)) + self.play(Write(y_xsquare_lab)) + self.wait(1) + + self.play(ShowCreation(y_1)) + self.play(Write(y_1_lab)) + self.wait(1) + + self.play(ShowCreation(y_2minusx)) + self.play(Write(y_2minusx_lab)) + self.wait(1) + + + + \ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file3_limit_approach_point_3d.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file3_limit_approach_point_3d.py new file mode 100644 index 0000000..f1007a4 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file3_limit_approach_point_3d.py @@ -0,0 +1,152 @@ +from manimlib.imports import * + +class Limit(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + + text3d = TextMobject(r"$f(x,y) = \frac{x - y}{x - 1}$") + self.add_fixed_in_frame_mobjects(text3d) + + text3d.to_corner(UL) + + text3d.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE) + + self.play(Write(text3d)) + self.wait(1) + + limit_func = ParametricSurface( + lambda u, v: np.array([ + 3*np.sin(u)*np.cos(v), + 3*np.sin(u)*np.sin(v), + (3*np.sin(u)*np.cos(v) - 3*np.sin(u)*np.sin(v))/2*(3*np.sin(u)*np.cos(v) - 1) + ]),u_min=0,u_max=PI,v_min=0,v_max=2*PI, color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1, + resolution=(15, 32)).scale(1) + + limit_y_x =ParametricFunction( + lambda u : np.array([ + u, + u, + 0 + ]),color=GREEN_D,t_min=-3,t_max=3, + ) + + limit_y_1 =ParametricFunction( + lambda u : np.array([ + u, + 1, + 1/2 + ]),color=BLUE_D,t_min=-3,t_max=3, + ) + + limit_y_x_2 =ParametricFunction( + lambda u : np.array([ + u, + u*u, + (u - u*u)/2*(u - 1) + ]),color=RED_D,t_min=-3,t_max=3, + ) + + limit_y_2_x =ParametricFunction( + lambda u : np.array([ + u, + 2 - u, + 1 + ]),color=YELLOW_D,t_min=-3,t_max=3, + ) + + plane_y_x = Polygon(np.array([-3,-3,-3]),np.array([3,3,-3]),np.array([3,3,3]),np.array([-3,-3,3]),np.array([-3,-3,-3]), color = GREEN_C, fill_color = GREEN_C, fill_opacity = 0.1) + plane_y_x_text = TextMobject(r"$y = x$", color = GREEN_C).move_to(np.array([5,0,3])) + + plane_y_1 = Polygon(np.array([-3,1,-3]),np.array([3,1,-3]),np.array([3,1,3]),np.array([-3,1,3]),np.array([-3,1,-3]), color = BLUE_C, fill_color = BLUE_C, fill_opacity = 0.1) + plane_y_1_text = TextMobject(r"$y = 1$", color = BLUE_C).move_to(np.array([5,0,2.5])) + + + #Creating plane y = x^2 + ###### + y_x_2 = [] + y_x_2.append(np.array([2, 4, -3])) + y_x_2.append(np.array([2, 4, 3])) + y_x_2_1 = [np.array([i, i*i, 3]) for i in np.arange(1.9,-2.1, -0.1)] + + y_x_2 = y_x_2 + y_x_2_1 + + y_x_2.append(np.array([-2, 4, 3])) + y_x_2.append(np.array([-2, 4, -3])) + + y_x_2_2 = [np.array([i, i*i, -3]) for i in np.arange(-2,2.1, 0.1)] + + y_x_2 = y_x_2 + y_x_2_2 + #y_x_2.append(np.array([-3, 9, 0])) + + plane_y_x_2 = Polygon(*y_x_2, color = RED_C, fill_color = RED_C, fill_opacity = 0.1) + plane_y_x_2_text = TextMobject(r"$y = x^2$", color = RED_C).move_to(np.array([5,0,2])) + + ###### + + plane_y_2_x = Polygon(np.array([-3,5,-3]),np.array([3,-1,-3]),np.array([3,-1,3]),np.array([-3,5,3]),np.array([-3,5,-3]), color = YELLOW_C, fill_color = YELLOW_C, fill_opacity = 0.1) + plane_y_2_x_text = TextMobject(r"$y = 2 - x$", color = YELLOW_C).move_to(np.array([5,0,1.5])) + + line_1_1 = Line(np.array([1,1,-3]), np.array([1,1,3]), color = PINK) + + point = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).move_to(np.array([1,1,0])) + point_text = TextMobject(r"$(1,1,0)$", color = WHITE).scale(0.7).move_to(np.array([1.8,1,0])) + + + + + self.set_camera_orientation(phi=70 * DEGREES, theta = -95*DEGREES) + + + self.add(axes) + + axis = TextMobject(r"X",r"Y",r"Z") + axis[0].move_to(6*RIGHT) + axis[1].move_to(6*UP) + axis[2].move_to(3.7*UP) + + self.add_fixed_in_frame_mobjects(axis[2]) + self.add_fixed_orientation_mobjects(axis[0]) + self.add_fixed_orientation_mobjects(axis[1]) + + self.play(ShowCreation(limit_func)) + self.wait(2) + + self.play(ShowCreation(plane_y_x)) + self.add_fixed_orientation_mobjects(plane_y_x_text) + self.play(ShowCreation(limit_y_x)) + self.wait() + + self.play(ShowCreation(plane_y_1)) + self.add_fixed_orientation_mobjects(plane_y_1_text) + self.play(ShowCreation(limit_y_1)) + self.wait() + + self.play(ShowCreation(plane_y_x_2)) + self.add_fixed_orientation_mobjects(plane_y_x_2_text) + self.play(ShowCreation(limit_y_x_2)) + self.wait() + + self.play(ShowCreation(plane_y_2_x)) + self.add_fixed_orientation_mobjects(plane_y_2_x_text) + self.play(ShowCreation(limit_y_2_x)) + self.wait() + + self.play(ShowCreation(line_1_1)) + self.wait() + + self.play(ShowCreation(point)) + self.add_fixed_orientation_mobjects(point_text) + self.wait() + + self.play(FadeOut(plane_y_x_text), FadeOut(plane_y_1_text), FadeOut(plane_y_x_2_text), FadeOut(plane_y_2_x_text)) + + self.move_camera(phi=0* DEGREES,theta=-95*DEGREES) + self.wait(2) + self.play(FadeOut(plane_y_x), FadeOut(plane_y_1), FadeOut(plane_y_x_2), FadeOut(plane_y_2_x)) + self.wait(3) + + self.move_camera(phi=75* DEGREES,theta=-95*DEGREES) + self.wait(3) + + + \ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file4_limit_different_point.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file4_limit_different_point.py new file mode 100644 index 0000000..0a43def --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file4_limit_different_point.py @@ -0,0 +1,115 @@ +from manimlib.imports import * + +class DifferentPoint(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + + text3d = TextMobject(r"$f(x,y) = \frac{x^2 - y^2}{x^2 + y^2}$") + self.add_fixed_in_frame_mobjects(text3d) + + text3d.to_corner(UL) + + text3d.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE) + + self.play(Write(text3d)) + self.wait(1) + + limit_func = ParametricSurface( + lambda u, v: np.array([ + 3*np.sin(u)*np.cos(v), + 3*np.sin(u)*np.sin(v), + (np.cos(v)*np.cos(v) - np.sin(v)*np.sin(v)) + ]),u_min=0,u_max=PI,v_min=0,v_max=2*PI,checkerboard_colors=[YELLOW_C, YELLOW_E], + resolution=(15, 32)).scale(1) + + limit_func_copy1 = limit_func.copy() + limit_func_copy2 = limit_func.copy() + + limit_func_x = ParametricSurface( + lambda u, v: np.array([ + 3*np.sin(u)*np.cos(v), + 3*np.sin(u)*np.sin(v), + (np.cos(v)*np.cos(v) - np.sin(v)*np.sin(v)) + ]),u_min=0,u_max=PI,v_min=PI,v_max=2*PI,checkerboard_colors=[YELLOW_C, YELLOW_E], + resolution=(15, 32)).scale(1) + + limit_func_y = ParametricSurface( + lambda u, v: np.array([ + 3*np.sin(u)*np.cos(v), + 3*np.sin(u)*np.sin(v), + (np.cos(v)*np.cos(v) - np.sin(v)*np.sin(v)) + ]),u_min=0,u_max=PI,v_min=PI/2,v_max=3*PI/2,checkerboard_colors=[YELLOW_C, YELLOW_E], + resolution=(15, 32)).scale(1) + + limit_x =ParametricFunction( + lambda u : np.array([ + u, + 0, + 1 + ]),color="#006400",t_min=-3,t_max=3, + ) + + limit_y =ParametricFunction( + lambda u : np.array([ + 0, + u, + -1 + ]),color="#000080",t_min=-3,t_max=3, + ) + + plane_x = Polygon(np.array([-3,0,-2]),np.array([3,0,-2]),np.array([3,0,2]),np.array([-3,0,2]),np.array([-3,0,-2]), color = GREEN, fill_color = GREEN, fill_opacity = 0.2) + plane_x_text = TextMobject(r"$y = 0$", color = GREEN_C).move_to(1.7*UP + 3.8*RIGHT) + + plane_y = Polygon(np.array([0,-3,-2]),np.array([0,3,-2]),np.array([0,3,2]),np.array([0,-3,2]),np.array([0,-3,-2]), color = BLUE, fill_color = BLUE, fill_opacity = 0.2) + plane_y_text = TextMobject(r"$x = 0$", color = BLUE_C).move_to(1.7*UP + 3.8*RIGHT) + + origin_x = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).move_to(np.array([0,0,0])) + origin_x_text = TextMobject(r"$(0,0,0)$", color = RED_C).scale(0.7).move_to(np.array([-0.6,0,-0.5])) + + origin_y = Polygon(*[np.array([0,0.05*np.cos(i*DEGREES),0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).move_to(np.array([0,0,0])) + origin_y_text = TextMobject(r"$(0,0,0)$", color = RED_C).scale(0.7).move_to(np.array([0,-0.6,-0.5])) + + self.set_camera_orientation(phi=80 * DEGREES, theta = 0*DEGREES) + + + self.add(axes) + + axis = TextMobject(r"X",r"Y",r"Z") + axis[0].move_to(6*RIGHT) + axis[1].move_to(6*UP) + axis[2].move_to(3.7*UP) + + self.add_fixed_in_frame_mobjects(axis[2]) + self.add_fixed_orientation_mobjects(axis[0]) + self.add_fixed_orientation_mobjects(axis[1]) + + self.play(ShowCreation(limit_func)) + + self.move_camera(phi=80* DEGREES,theta=105*DEGREES) + + self.play(ShowCreation(plane_x)) + self.add_fixed_in_frame_mobjects(plane_x_text) + self.wait() + self.play(ReplacementTransform(limit_func, limit_func_x)) + self.play(FadeOut(plane_x), FadeOut(plane_x_text), ShowCreation(origin_x)) + self.add_fixed_orientation_mobjects(origin_x_text) + self.play(ShowCreation(limit_x)) + + self.move_camera(phi=80* DEGREES,theta=15*DEGREES) + self.wait(3) + + self.play(FadeOut(origin_x), FadeOut(origin_x_text), FadeOut(limit_x), ReplacementTransform(limit_func_x, limit_func_copy1)) + self.play(ShowCreation(plane_y)) + self.add_fixed_in_frame_mobjects(plane_y_text) + self.wait() + self.play(ReplacementTransform(limit_func_copy1, limit_func_y)) + self.play(FadeOut(plane_y), FadeOut(plane_y_text), ShowCreation(origin_y)) + self.add_fixed_orientation_mobjects(origin_y_text) + self.play(ShowCreation(limit_y)) + + self.move_camera(phi=80* DEGREES,theta=75*DEGREES) + self.wait(3) + + self.play(FadeOut(origin_y), FadeOut(origin_y_text), FadeOut(limit_y), ReplacementTransform(limit_func_y, limit_func_copy2)) + self.wait(2) + \ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file5_continuity_func.py b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file5_continuity_func.py new file mode 100644 index 0000000..99159a4 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/file5_continuity_func.py @@ -0,0 +1,115 @@ +from manimlib.imports import * + +class Continuity(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + + text3d = TextMobject(r"$f(x,y) = \frac{3x^2y}{x^2 + y^2}$") + self.add_fixed_in_frame_mobjects(text3d) + + text3d.to_corner(UL) + + text3d.set_color_by_gradient(RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE) + + self.play(Write(text3d)) + self.wait(1) + + + continuity_func = ParametricSurface( + lambda u, v: np.array([ + 3*np.sin(u)*np.cos(v), + 3*np.sin(u)*np.sin(v), + 9*np.sin(u)*np.cos(v)*np.cos(v)*np.sin(v) + ]),u_min=0,u_max=PI,v_min=0,v_max=2*PI,checkerboard_colors=[YELLOW_C, YELLOW_E], + resolution=(15, 32)).scale(1) + + continuity_func_copy1 = continuity_func.copy() + continuity_func_copy2 = continuity_func.copy() + + continuity_func_x = ParametricSurface( + lambda u, v: np.array([ + 3*np.sin(u)*np.cos(v), + 3*np.sin(u)*np.sin(v), + 9*np.sin(u)*np.cos(v)*np.cos(v)*np.sin(v) + ]),u_min=0,u_max=PI,v_min=PI,v_max=2*PI,checkerboard_colors=[YELLOW_C, YELLOW_E], + resolution=(15, 32)).scale(1) + + continuity_func_y = ParametricSurface( + lambda u, v: np.array([ + 3*np.sin(u)*np.cos(v), + 3*np.sin(u)*np.sin(v), + 9*np.sin(u)*np.cos(v)*np.cos(v)*np.sin(v) + ]),u_min=0,u_max=PI,v_min=PI/2,v_max=3*PI/2,checkerboard_colors=[YELLOW_C, YELLOW_E], + resolution=(15, 32)).scale(1) + + continuity_x =ParametricFunction( + lambda u : np.array([ + u, + 0, + 0 + ]),color="#006400",t_min=-3,t_max=3, + ) + + continuity_y =ParametricFunction( + lambda u : np.array([ + 0, + u, + 0 + ]),color="#000080",t_min=-3,t_max=3, + ) + + plane_x = Polygon(np.array([-3,0,-3]),np.array([3,0,-3]),np.array([3,0,3]),np.array([-3,0,3]),np.array([-3,0,-3]), color = GREEN, fill_color = GREEN, fill_opacity = 0.2) + plane_x_text = TextMobject(r"$y = 0$", color = GREEN_C).move_to(1.7*UP + 3.8*RIGHT) + + plane_y = Polygon(np.array([0,-3,-3]),np.array([0,3,-3]),np.array([0,3,3]),np.array([0,-3,3]),np.array([0,-3,-3]), color = BLUE, fill_color = BLUE, fill_opacity = 0.2) + plane_y_text = TextMobject(r"$x = 0$", color = BLUE_C).move_to(1.7*UP + 3.8*RIGHT) + + origin_x = Polygon(*[np.array([0.05*np.cos(i*DEGREES),0,0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#000080", fill_color = "#000080", fill_opacity = 1).move_to(np.array([0,0,0])) + origin_x_text = TextMobject(r"$(0,0,0)$", color = RED_C).scale(0.7).move_to(np.array([-0.6,0,-0.5])) + + origin_y = Polygon(*[np.array([0,0.05*np.cos(i*DEGREES),0.05*np.sin(i*DEGREES)]) for i in range(361)], color = "#006400", fill_color = "#006400", fill_opacity = 1).move_to(np.array([0,0,0])) + origin_y_text = TextMobject(r"$(0,0,0)$", color = RED_C).scale(0.7).move_to(np.array([0,-0.6,-0.5])) + + self.set_camera_orientation(phi=80 * DEGREES, theta = 0*DEGREES) + + + self.add(axes) + + axis = TextMobject(r"X",r"Y",r"Z") + axis[0].move_to(6*RIGHT) + axis[1].move_to(6*UP) + axis[2].move_to(3.7*UP) + + self.add_fixed_in_frame_mobjects(axis[2]) + self.add_fixed_orientation_mobjects(axis[0]) + self.add_fixed_orientation_mobjects(axis[1]) + + self.play(ShowCreation(continuity_func)) + + self.move_camera(phi=80* DEGREES,theta=105*DEGREES) + + self.play(ShowCreation(plane_x)) + self.add_fixed_in_frame_mobjects(plane_x_text) + self.wait() + self.play(ReplacementTransform(continuity_func, continuity_func_x)) + self.play(FadeOut(plane_x), FadeOut(plane_x_text)) + self.play(ShowCreation(continuity_x), ShowCreation(origin_x)) + self.add_fixed_orientation_mobjects(origin_x_text) + + self.move_camera(phi=80* DEGREES,theta=15*DEGREES) + self.wait(3) + + self.play(FadeOut(origin_x), FadeOut(origin_x_text), FadeOut(continuity_x), ReplacementTransform(continuity_func_x, continuity_func_copy1)) + self.play(ShowCreation(plane_y)) + self.add_fixed_in_frame_mobjects(plane_y_text) + self.wait() + self.play(ReplacementTransform(continuity_func_copy1, continuity_func_y)) + self.play(FadeOut(plane_y), FadeOut(plane_y_text)) + self.play(ShowCreation(continuity_y), ShowCreation(origin_y)) + self.add_fixed_orientation_mobjects(origin_y_text) + + self.move_camera(phi=80* DEGREES,theta=75*DEGREES) + self.wait(3) + + self.play(FadeOut(origin_y), FadeOut(origin_y_text), FadeOut(continuity_y), ReplacementTransform(continuity_func_y, continuity_func_copy2)) + self.wait(2) \ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file1_epsilon_delta_defn.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file1_epsilon_delta_defn.gif new file mode 100644 index 0000000..2378bcf Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file1_epsilon_delta_defn.gif differ diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file2_limit_approach_point.gif 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