diff options
Diffstat (limited to 'FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions')
46 files changed, 3543 insertions, 28 deletions
diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/README.md b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/README.md index a321caf..4de6c1d 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/README.md +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/README.md @@ -1 +1,13 @@ FSF2020--Somnath Pandit
+
+# **Topics:**
+
+## Double Integral
+
+## Fubini's Theorem
+
+## Line Integrals
+
+## Fundamental Theorem of Line integrals
+
+## Vector Fields
diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/README.md b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/README.md new file mode 100644 index 0000000..5fa2146 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/README.md @@ -0,0 +1,21 @@ +**file1_area_under_func** + + +**file2_volume_under_surface** + + +**file3_y_limit_dependent_on_x** + + +**file4_non_rect_region** + + +**file5_elementary_area** + + +**file6_doing_integration** + + +**file7_int_process_of_example** + + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/area_Under_func.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file1_area_under_func.gif Binary files differindex 223218b..223218b 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/area_Under_func.gif +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file1_area_under_func.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/area_under_func.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file1_area_under_func.py index 773840c..773840c 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/area_under_func.py +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file1_area_under_func.py diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file2_volume_under_surface.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file2_volume_under_surface.gif Binary files differnew file mode 100644 index 0000000..1455573 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file2_volume_under_surface.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/surface.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file2_volume_under_surface.py index a794f46..38d41c6 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/surface.py +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file2_volume_under_surface.py @@ -5,11 +5,11 @@ class SurfacesAnimation(ThreeDScene): CONFIG = { "axes_config": { "x_min": 0, - "x_max": 8, + "x_max": 7, "y_min": 0, - "y_max": 8, + "y_max": 7, "z_min": 0, - "z_max": 6, + "z_max": 5, "a":1 ,"b": 6, "c":2 , "d":6, "axes_shift":-3*OUT + 5*LEFT, "x_axis_config": { @@ -46,13 +46,24 @@ class SurfacesAnimation(ThreeDScene): self.setup_axes() self.set_camera_orientation(distance=35, phi=80 * DEGREES, - theta=-80 * DEGREES, + theta=-100 * DEGREES, ) - fn_text=TextMobject("$z=f(x,y)$").set_color(PINK) + fn_text=TextMobject( + "$z=f(x,y)$", + color=PINK, + stroke_width=1.5 + ) self.add_fixed_in_frame_mobjects(fn_text) fn_text.to_edge(TOP,buff=MED_SMALL_BUFF) + riemann_sum_text=TextMobject( + r"The volume approximated as\\ sum of cuboids", + color=BLUE, + stroke_width=1.5 + ) + riemann_sum_text.to_corner(UR,buff=.2) + R=TextMobject("R").set_color(BLACK).scale(3) R.move_to(self.axes.input_plane,IN) self.add(R) @@ -70,13 +81,54 @@ class SurfacesAnimation(ThreeDScene): ) - self.begin_ambient_camera_rotation(rate=0.07) + self.begin_ambient_camera_rotation(rate=0.06) self.play(Write(surface)) - # self.play(LaggedStart(ShowCreation(surface))) + # self.add(surface) self.get_lines() - # self.play(FadeIn(self.axes.input_plane)) + self.wait(1) + self.add_fixed_in_frame_mobjects(riemann_sum_text) + self.play(Write(riemann_sum_text)) + + cuboids1=self.show_the_riemmann_sum( + lambda x,y : np.array([x,y,self.Func(x,y)]), + fill_opacity=1, + dl=.5, + start_color=BLUE, + end_color=BLUE_E, + ) + self.play(ShowCreation(cuboids1),run_time=5) + self.play(FadeOut(surface)) + + cuboids2=self.show_the_riemmann_sum( + lambda x,y : np.array([x,y,self.Func(x,y)]), + fill_opacity=1, + dl=.25, + start_color=BLUE, + end_color=BLUE_E, + ) + self.play(ReplacementTransform( + cuboids1, + cuboids2 + )) + + cuboids3=self.show_the_riemmann_sum( + lambda x,y : np.array([x,y,self.Func(x,y)]), + fill_opacity=1, + dl=.1, + start_color=BLUE, + end_color=BLUE_E, + stroke_width=.1, + ) + self.play( + FadeOut(cuboids2), + ShowCreation(cuboids3), + ) + self.wait(3) + + + def get_surface(self,axes, func, **kwargs): config = { @@ -101,9 +153,6 @@ class SurfacesAnimation(ThreeDScene): def get_lines(self): axes = self.axes - labels=[axes.x_axis.n2p(axes.a), axes.x_axis.n2p(axes.b), axes.y_axis.n2p(axes.c), - axes.y_axis.n2p(axes.d)] - surface_corners=[] for x,y,z in self.region_corners: @@ -112,23 +161,89 @@ class SurfacesAnimation(ThreeDScene): lines=VGroup() for start , end in zip(surface_corners, self.region_corners): - lines.add(self.draw_lines(start,end,"RED")) + lines.add(self.draw_lines(start,end,"#9CDCEB")) + + labels=[ + (axes.a,0,0), + (axes.b,0,0), + (0,axes.d,0), + (0,axes.c,0) + ] + self.region_corners[-1]=self.region_corners[0] + for start , end in zip(labels,self.region_corners): + lines.add(self.draw_lines(start,end,"WHITE")) - for start , end in zip(labels, - self.region_corners): - # lines.add(self.draw_lines(start,end,"BLUE")) - # print (start,end) - pass + # self.add(lines) self.play(ShowCreation(lines)) - + def draw_lines(self,start,end,color): start=self.axes.c2p(*start) end=self.axes.c2p(*end) line=DashedLine(start,end,color=color) return line + + + def show_the_riemmann_sum( + self, + surface, + x_min=None, + x_max=None, + y_min=None, + y_max=None, + dl=.5, + stroke_width=.5, + stroke_color=BLACK, + fill_opacity=1, + start_color=None, + end_color=None, + ): + x_min = x_min if x_min is not None else self.axes.a + x_max = x_max if x_max is not None else self.axes.b + y_min = y_min if y_min is not None else self.axes.c + y_max = y_max if y_max is not None else self.axes.d + + if start_color is None: + start_color = BLUE + if end_color is None: + end_color = BLUE + + cuboids = VGroup() + x_range = np.arange(x_min, x_max, dl) + y_range = np.arange(y_min, y_max, dl) + colors = color_gradient([start_color, end_color], len(x_range)) + for x, color in zip(x_range, colors): + for y in y_range: + sample_base = np.array([x ,y ,0]) + sample_base_dl = np.array([x + dl, y + dl,0]) + sample_input = np.array([x +0.5*dl, y +0.5*dl,0]) + + base_point = self.axes.c2p(*sample_base) + base_dx_point = self.axes.c2p(*sample_base_dl) + + surface_val= surface(*sample_input[:2]) + surface_point = self.axes.c2p(*surface_val) + + points = VGroup(*list(map(VectorizedPoint, [ + base_point, + surface_point, + base_dx_point + ]))) + + # self.add(points) + cuboid = Prism(dimensions=[dl,dl,surface_val[-1]]) + cuboid.replace(points, stretch=True) + + cuboid.set_fill(color, opacity=fill_opacity) + cuboid.set_stroke(stroke_color, width=stroke_width) + cuboids.add(cuboid) + + return cuboids + +#------------------------------------------------------- + #customize 3d axes def get_three_d_axes(self, include_labels=True, include_numbers=True, **kwargs): config = dict(self.axes_config) config.update(kwargs) @@ -165,7 +280,7 @@ class SurfacesAnimation(ThreeDScene): axes.input_plane = input_plane self.region_corners=[ - input_plane.get_corner(pos) for pos in (DL,DR,UR,UL)] + input_plane.get_corner(pos) for pos in (DL,DR,UL,UR)] return axes @@ -232,5 +347,3 @@ class SurfacesAnimation(ThreeDScene): return axes - -#uploaded by Somnath Pandit.FSF2020_Double_Integral diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file3_y_limit_dependent_on_x.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file3_y_limit_dependent_on_x.gif Binary files differnew file mode 100644 index 0000000..dcfacb6 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file3_y_limit_dependent_on_x.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/y_limit_dependent_on_x.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file3_y_limit_dependent_on_x.py index 4894ebf..f755495 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/y_limit_dependent_on_x.py +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file3_y_limit_dependent_on_x.py @@ -29,7 +29,7 @@ class YlimitXdependent(GraphScene): line_eqn=TextMobject("2x+y=2").move_to(self.graph_origin+.8*X+Y).rotate(np.arctan(-2)) self.line=line - caption=TextMobject(r"See the value of $y$ \\ is changing with $x$").move_to(self.graph_origin+1.2*X+1.8*Y) + caption=TextMobject(r"The value of $y$ is\\ changing with $x$").move_to(self.graph_origin+1.2*X+1.8*Y) self.play(ShowCreation(line),Write(line_eqn)) # self.show_area() self.show_rects() diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/non_rect_region.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file4_non_rect_region.gif Binary files differindex c8e7c8c..c8e7c8c 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/non_rect_region.gif +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file4_non_rect_region.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/non_rect_region.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file4_non_rect_region.py index 793a000..793a000 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/non_rect_region.py +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file4_non_rect_region.py diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/elementary_area.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file5_elementary_area.gif Binary files differindex 5c9ac03..5c9ac03 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/elementary_area.gif +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file5_elementary_area.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/elementary_area.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file5_elementary_area.py index 362b6f8..362b6f8 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/elementary_area.py +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file5_elementary_area.py diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file6_doing_integration.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file6_doing_integration.gif Binary files differnew file mode 100644 index 0000000..7a9271b --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file6_doing_integration.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file6_doing_integration.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file6_doing_integration.py new file mode 100644 index 0000000..5a8cec0 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file6_doing_integration.py @@ -0,0 +1,355 @@ +from manimlib.imports import * + +class IntegrationProcess(SpecialThreeDScene): + + CONFIG = { + "axes_config": { + "x_min": 0, + "x_max": 7, + "y_min": 0, + "y_max": 7, + "z_min": 0, + "z_max": 4, + "a":1 ,"b": 6, "c":2 , "d":6, + "axes_shift":-3*OUT, + "x_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "y_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "z_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "num_axis_pieces": 1, + }, + "default_graph_style": { + "stroke_width": 2, + "stroke_color": WHITE, + }, + "default_surface_config": { + "fill_opacity": 0.5, + "checkerboard_colors": [LIGHT_GREY], + "stroke_width": 0.5, + "stroke_color": WHITE, + "stroke_opacity": 0.5, + }, + "Func": lambda x,y: 2+y/4+np.cos(x/1.4) + } + + + def construct(self): + + self.setup_axes() + axes=self.axes + + self.camera.frame_center.shift(axes.c2p(3,4,1.7)) + self.set_camera_orientation(distance=35, + phi= 80 * DEGREES, + theta= -80 * DEGREES, + gamma = 0 * DEGREES + ) + + fn_text=TextMobject("$z=f(x,y)$").set_color(PINK) + self.add_fixed_in_frame_mobjects(fn_text) + + + R=TextMobject("R").set_color(BLACK).scale(3) + R.move_to(axes.input_plane,IN) + self.add(R) + + # get the surface + surface= self.get_surface( + axes, lambda x , y: + self.Func(x,y) + ) + surface.set_style( + fill_opacity=.65, + fill_color=PINK, + stroke_width=0.8, + stroke_color=WHITE, + ) + fn_text.next_to(surface,UP,buff=MED_LARGE_BUFF) + slice_curve=(self.get_y_slice_graph( + axes,self.Func,5,color=YELLOW)) + + + self.begin_ambient_camera_rotation(rate=0.08) + # self.play(Write(surface)) + self.add(surface) + + self.get_lines() + + self.show_process(axes) + + self.wait(3) + + + + def show_process(self,axes): + y_tracker = ValueTracker(axes.c) + self.y_tracker=y_tracker + y=y_tracker.get_value + + graph = always_redraw( + lambda: self.get_y_slice_graph( + axes, self.Func, y(), + stroke_color=YELLOW, + stroke_width=4, + ) + ) + graph.suspend_updating() + + + plane = always_redraw(lambda: Polygon( + *[ + axes.c2p(x,y(),self.Func(x,y())) + for x in np.arange(axes.a,axes.b,0.01) + ], + *[ + axes.c2p(x, y(), 0) + for x in [ axes.b, axes.a,] + ], + stroke_width=2, + fill_color=BLUE_D, + fill_opacity=.4, + )) + + plane_side1 = always_redraw(lambda: Polygon( + *[ + axes.c2p(axes.a,y,self.Func(axes.a,y)) + for y in np.arange(axes.c,y(),0.01) + ], + *[ + axes.c2p(axes.a, y, 0) + for y in [ y(),axes.c, ] + ], + stroke_width=2.5, + fill_color=BLUE_C, + fill_opacity=.2, + )) + plane_side2 = always_redraw(lambda: Polygon( + *[ + axes.c2p(axes.b,y,self.Func(axes.b,y)) + for y in np.arange(axes.c,y(),0.01) + ], + *[ + axes.c2p(axes.b, y, 0) + for y in [y(),axes.c,] + ], + stroke_width=2.5, + fill_color=BLUE_E, + fill_opacity=.45, + )) + plane.suspend_updating() + plane_side1.suspend_updating() + plane_side2.suspend_updating() + + self.play(Write(VGroup(graph,plane)),run_time=2) + self.add(plane.copy(),plane_side1,plane_side2) + + + plane_side1.resume_updating() + plane_side2.resume_updating() + + self.move_camera( + distance=30, + phi= 85 * DEGREES, + theta= -10 * DEGREES, + run_time=1.5 + ) + self.play( + ApplyMethod( + y_tracker.set_value, axes.d, + rate_func=linear, + run_time=6, + ) + ) + plane.suspend_updating() + plane_side1.suspend_updating() + plane_side2.suspend_updating() + + + + def get_y_slice_graph(self, axes, func, y, **kwargs): + config = dict() + config.update(self.default_graph_style) + config.update({ + "t_min": axes.a, + "t_max": axes.b, + }) + config.update(kwargs) + slice_curve=ParametricFunction( + lambda x: axes.c2p( + x, y, func(x, y) + ), + **config, + ) + return slice_curve + + + def get_surface(self,axes, func, **kwargs): + config = { + "u_min": axes.a, + "u_max": axes.b, + "v_min": axes.c, + "v_max": axes.d, + "resolution": ( + (axes.y_max - axes.y_min) // axes.y_axis.tick_frequency, + (axes.x_max - axes.x_min) // axes.x_axis.tick_frequency, + ), + } + + config.update(self.default_surface_config) + config.update(kwargs) + return ParametricSurface( + lambda x,y : axes.c2p( + x, y, func(x, y) + ), + **config + ) + + def get_lines(self): + axes = self.axes + + surface_corners=[] + for x,y,z in self.region_corners: + surface_corners.append([x,y,self.Func(x,y)]) + + lines=VGroup() + for start , end in zip(surface_corners, + self.region_corners): + lines.add(self.draw_lines(start,end,"RED")) + + labels=[ + (axes.a,0,0), + (axes.b,0,0), + (0,axes.d,0), + (0,axes.c,0) + ] + self.region_corners[-1]=self.region_corners[0] + for start , end in zip(labels, + self.region_corners): + lines.add(self.draw_lines(start,end,"WHITE")) + self.add(lines) + + + def draw_lines(self,start,end,color): + start=self.axes.c2p(*start) + end=self.axes.c2p(*end) + line=DashedLine(start,end,color=color) + + return line + + +#------------------------------------------------------------ + #customize 3d axes + def get_three_d_axes(self, include_labels=True, include_numbers=True, **kwargs): + config = dict(self.axes_config) + config.update(kwargs) + axes = ThreeDAxes(**config) + axes.set_stroke(width=2) + + if include_numbers: + self.add_axes_numbers(axes) + + if include_labels: + self.add_axes_labels(axes) + + # Adjust axis orientation + axes.x_axis.rotate( + 90 * DEGREES, RIGHT, + about_point=axes.c2p(0, 0, 0), + ) + axes.y_axis.rotate( + 90 * DEGREES, UP, + about_point=axes.c2p(0, 0, 0), + ) + + # Add xy-plane + input_plane = self.get_surface( + axes, lambda x, t: 0 + ) + input_plane.set_style( + fill_opacity=0.5, + fill_color=TEAL, + stroke_width=0, + stroke_color=WHITE, + ) + + axes.input_plane = input_plane + + self.region_corners=[ + input_plane.get_corner(pos) for pos in (DL,DR,UL,UR)] + + return axes + + + def setup_axes(self): + axes = self.get_three_d_axes(include_labels=True) + axes.add(axes.input_plane) + axes.scale(1) + # axes.center() + axes.shift(axes.axes_shift) + + self.add(axes) + + self.axes = axes + + def add_axes_numbers(self, axes): + x_axis = axes.x_axis + y_axis = axes.y_axis + tex_vals_x = [ + ("a", axes.a), + ("b", axes.b), + ] + tex_vals_y=[ + ("c", axes.c), + ("d", axes.d) + ] + x_labels = VGroup() + y_labels = VGroup() + for tex, val in tex_vals_x: + label = TexMobject(tex) + label.scale(1) + label.next_to(x_axis.n2p(val), DOWN) + x_labels.add(label) + x_axis.add(x_labels) + x_axis.numbers = x_labels + + for tex, val in tex_vals_y: + label = TexMobject(tex) + label.scale(1.5) + label.next_to(y_axis.n2p(val), LEFT) + label.rotate(90 * DEGREES) + y_labels.add(label) + + y_axis.add(y_labels) + y_axis.numbers = y_labels + + return axes + + def add_axes_labels(self, axes): + x_label = TexMobject("x") + x_label.next_to(axes.x_axis.get_end(), RIGHT) + axes.x_axis.label = x_label + + y_label = TextMobject("y") + y_label.rotate(90 * DEGREES, OUT) + y_label.next_to(axes.y_axis.get_end(), UP) + axes.y_axis.label = y_label + + z_label = TextMobject("z") + z_label.rotate(90 * DEGREES, RIGHT) + z_label.next_to(axes.z_axis.get_zenith(), RIGHT) + axes.z_axis.label = z_label + for axis in axes: + axis.add(axis.label) + return axes + + + + #uploaded by Somnath Pandit.FSF2020_Double_Integral diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file7_int_process_of_example.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file7_int_process_of_example.gif Binary files differnew file mode 100644 index 0000000..9fbdde8 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file7_int_process_of_example.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file7_int_process_of_example.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file7_int_process_of_example.py new file mode 100644 index 0000000..f733761 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/file7_int_process_of_example.py @@ -0,0 +1,366 @@ +from manimlib.imports import * + +class IntegrationProcess(SpecialThreeDScene): + + CONFIG = { + "axes_config": { + "x_min": 0, + "x_max": 5, + "y_min": 0, + "y_max": 7, + "z_min": 0, + "z_max": 3, + "a":0 ,"b":4 , "c":0 , "d":6, + "axes_shift":1.5*IN+2*LEFT+4*DOWN, + "x_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "y_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "z_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "num_axis_pieces": 1, + }, + "default_graph_style": { + "stroke_width": 2, + "stroke_color": WHITE, + }, + "default_surface_config": { + "fill_opacity": 0.5, + "checkerboard_colors": [LIGHT_GREY], + "stroke_width": 0.5, + "stroke_color": WHITE, + "stroke_opacity": 0.5, + }, + "Func": lambda x,y: 2*(1+(x+y)/10) + } + + + def construct(self): + + self.setup_axes() + axes=self.axes + self.set_camera_orientation(#distance=35, + phi=60 * DEGREES, + theta=10 * DEGREES, + ) + + fn_text=TextMobject("$z=(1+x+y)$").set_color(PINK) + self.add_fixed_in_frame_mobjects(fn_text) + fn_text.to_edge(TOP,buff=.1) + self.fn_text=fn_text + + R=TextMobject("R").set_color(BLACK).scale(3).rotate(PI/2) + R.move_to(axes.input_plane,IN) + self.add(R) + + #get the surface + surface= self.get_surface( + axes, lambda x , y: + self.Func(x,y) + ) + surface.set_style( + fill_opacity=0.75, + fill_color=PINK, + stroke_width=0.8, + stroke_color=WHITE, + ) + + slice_curve=(self.get_y_slice_graph( + axes,self.Func,5,color=YELLOW)) + + + self.begin_ambient_camera_rotation(rate=0.04) + # self.play(Write(surface)) + self.add(surface) + + self.get_lines() + + self.show_process(axes) + + self.wait() + + + + def show_process(self,axes): + y_tracker = ValueTracker(axes.c) + self.y_tracker=y_tracker + y=y_tracker.get_value + graph = always_redraw( + lambda: self.get_y_slice_graph( + axes, self.Func, y(), + stroke_color=YELLOW, + stroke_width=4, + ) + ) + graph.suspend_updating() + + plane = always_redraw(lambda: Polygon( + *[ + axes.c2p(x,y(),self.Func(x,y())) + for x in np.arange(axes.a,axes.b,0.01) + ], + *[ + axes.c2p(x, y(), 0) + for x in [ axes.b, axes.a,] + ], + stroke_width=0, + fill_color=BLUE_E, + fill_opacity=.65, + )) + plane_side1 = always_redraw(lambda: Polygon( + *[ + axes.c2p(axes.a,y,self.Func(axes.a,y)) + for y in np.arange(axes.c,y(),0.01) + ], + *[ + axes.c2p(axes.a, y, 0) + for y in [ y(),axes.c, ] + ], + stroke_width=2.5, + fill_color=BLUE_C, + fill_opacity=.2, + )) + plane_side2 = always_redraw(lambda: Polygon( + *[ + axes.c2p(axes.b,y,self.Func(axes.b,y)) + for y in np.arange(axes.c,y(),0.01) + ], + *[ + axes.c2p(axes.b, y, 0) + for y in [y(),axes.c,] + ], + stroke_width=2.5, + fill_color=BLUE_E, + fill_opacity=.45, + )) + plane.suspend_updating() + plane_side1.suspend_updating() + plane_side2.suspend_updating() + + first_x_text=TextMobject("First the $x$ integration..",color=YELLOW) + first_x_text.to_corner(UR,buff=1.1) + + x_func=TextMobject("$\\frac 3 2 + y$",color=BLUE) + '''x_func.next_to(self.fn_text,DOWN) + x_func.align_to(self.fn_text,LEFT)''' + x_func.to_edge(LEFT,buff=1) + + then_y_text=TextMobject("Then the $y$ integration..",color=YELLOW) + then_y_text.to_corner(UR,buff=1.1) + + self.add_fixed_in_frame_mobjects(first_x_text) + self.play(Write(first_x_text)) + self.add_fixed_in_frame_mobjects(x_func) + self.play( + Write(VGroup(graph,plane,x_func)), + run_time=3 + ) + + self.wait() + self.remove(first_x_text) + self.add_fixed_in_frame_mobjects(then_y_text) + self.play(Write(then_y_text)) + self.add(plane.copy(),plane_side1,plane_side2) + graph.resume_updating() + plane.resume_updating() + plane_side1.resume_updating() + plane_side2.resume_updating() + self.play( + ApplyMethod( + y_tracker.set_value, axes.d, + rate_func=linear, + run_time=6, + ) + ) + + graph.suspend_updating() + plane.suspend_updating() + plane_side1.suspend_updating() + plane_side2.suspend_updating() + + + def get_y_slice_graph(self, axes, func, y, **kwargs): + config = dict() + config.update(self.default_graph_style) + config.update({ + "t_min": axes.a, + "t_max": axes.b, + }) + config.update(kwargs) + slice_curve=ParametricFunction( + lambda x: axes.c2p( + x, y, func(x, y) + ), + **config, + ) + return slice_curve + + + def get_surface(self,axes, func, **kwargs): + config = { + "u_min": axes.a, + "u_max": axes.b, + "v_min": axes.c, + "v_max": axes.d, + "resolution": ( + (axes.y_max - axes.y_min) // axes.y_axis.tick_frequency, + (axes.x_max - axes.x_min) // axes.x_axis.tick_frequency, + ), + } + + config.update(self.default_surface_config) + config.update(kwargs) + return ParametricSurface( + lambda x,y : axes.c2p( + x, y, func(x, y) + ), + **config + ) + + def get_lines(self): + axes = self.axes + + surface_corners=[] + for x,y,z in self.region_corners: + surface_corners.append([x,y,self.Func(x,y)]) + + lines=VGroup() + for start , end in zip(surface_corners, + self.region_corners): + lines.add(self.draw_lines(start,end,"RED")) + + labels=[ + (axes.a,0,0), + (axes.b,0,0), + (0,axes.d,0), + (0,axes.c,0) + ] + self.region_corners[-1]=self.region_corners[0] + for start , end in zip(labels, + self.region_corners): + lines.add(self.draw_lines(start,end,"WHITE")) + self.add(lines) + + + def draw_lines(self,start,end,color): + start=self.axes.c2p(*start) + end=self.axes.c2p(*end) + line=DashedLine(start,end,color=color) + + return line + + +#------------------------------------------------------------ + #customize 3d axes + def get_three_d_axes(self, include_labels=True, include_numbers=True, **kwargs): + config = dict(self.axes_config) + config.update(kwargs) + axes = ThreeDAxes(**config) + axes.set_stroke(width=2) + + if include_numbers: + self.add_axes_numbers(axes) + + if include_labels: + self.add_axes_labels(axes) + + # Adjust axis orientation + axes.x_axis.rotate( + 90 * DEGREES, LEFT, + about_point=axes.c2p(0, 0, 0), + ) + axes.y_axis.rotate( + 90 * DEGREES, UP, + about_point=axes.c2p(0, 0, 0), + ) + + # Add xy-plane + input_plane = self.get_surface( + axes, lambda x, t: 0 + ) + input_plane.set_style( + fill_opacity=0.3, + fill_color=TEAL, + stroke_width=.2, + stroke_color=WHITE, + ) + + axes.input_plane = input_plane + + self.region_corners=[ + input_plane.get_corner(pos) for pos in (DL,DR,UL,UR)] + + return axes + + + def setup_axes(self): + axes = self.get_three_d_axes(include_labels=True) + axes.add(axes.input_plane) + axes.scale(1) + # axes.center() + axes.shift(axes.axes_shift) + + self.add(axes) + self.axes = axes + + def add_axes_numbers(self, axes): + x_axis = axes.x_axis + y_axis = axes.y_axis + tex_vals_x = [ + + ("1", axes.b), + ] + tex_vals_y=[ + + ("2", axes.d) + ] + x_labels = VGroup() + y_labels = VGroup() + for tex, val in tex_vals_x: + label = TexMobject(tex) + label.scale(1) + label.next_to(x_axis.n2p(val), DOWN) + label.rotate(180 * DEGREES) + x_labels.add(label) + x_axis.add(x_labels) + x_axis.numbers = x_labels + + for tex, val in tex_vals_y: + label = TexMobject(tex) + label.scale(1) + label.next_to(y_axis.n2p(val), LEFT) + label.rotate(90 * DEGREES) + y_labels.add(label) + + y_axis.add(y_labels) + y_axis.numbers = y_labels + + return axes + + def add_axes_labels(self, axes): + x_label = TexMobject("x") + x_label.next_to(axes.x_axis.get_end(), RIGHT) + axes.x_axis.label = x_label + + y_label = TextMobject("y") + y_label.rotate(90 * DEGREES, OUT) + y_label.next_to(axes.y_axis.get_end(), UP) + axes.y_axis.label = y_label + + z_label = TextMobject("z") + z_label.rotate(90 * DEGREES, LEFT) + z_label.next_to(axes.z_axis.get_zenith(), LEFT) + axes.z_axis.label = z_label + for axis in axes: + axis.add(axis.label) + return axes + + + + #uploaded by Somnath Pandit.FSF2020_Double_Integral diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/surface.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/surface.gif Binary files differdeleted file mode 100644 index ae23a7b..0000000 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/surface.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/y_limit_dependent_on_x.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/y_limit_dependent_on_x.gif Binary files differdeleted file mode 100644 index a2bfd9d..0000000 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/double-integrals/y_limit_dependent_on_x.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/README.md b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/README.md new file mode 100644 index 0000000..c1c6e8e --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/README.md @@ -0,0 +1,14 @@ + +**file1_surface1** + + +**file2_surface2** + + +**file3_iteration_methods** + + +**file4_curvy_limits** + + + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's_theorem/surface1.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file1_surface1.gif Binary files differindex 8c9fa0a..8c9fa0a 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's_theorem/surface1.gif +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file1_surface1.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's_theorem/surface1.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file1_surface1.py index a590a53..a590a53 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's_theorem/surface1.py +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file1_surface1.py diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file2_surface2.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file2_surface2.gif Binary files differnew file mode 100644 index 0000000..37c4b1d --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file2_surface2.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's_theorem/surface2.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file2_surface2.py index c998f3b..3160fdb 100644 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's_theorem/surface2.py +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file2_surface2.py @@ -63,7 +63,7 @@ class SurfacesAnimation(ThreeDScene): surface.set_style( fill_opacity=.5, fill_color=BLUE_E, - stroke_width=0.2, + stroke_width=0.4, stroke_color=WHITE, ) #get boundary curves @@ -85,17 +85,18 @@ class SurfacesAnimation(ThreeDScene): self.add(c1,c2,c1_label,c2_label) - self.begin_ambient_camera_rotation(rate=0.4) + self.begin_ambient_camera_rotation(rate=0.24) self.get_region(self.axes,c1,c2) self.play(Write(surface)) self.get_lines() - self.wait(1) + self.wait(3.5) self.stop_ambient_camera_rotation() + self.wait(.5) self.move_camera( distance=20, phi=10 * DEGREES, theta=80 * DEGREES, - run_time=2.5 + run_time=3 ) self.wait(2) @@ -128,7 +129,7 @@ class SurfacesAnimation(ThreeDScene): R=TextMobject("R").set_color(PINK).scale(2).rotate(180*DEGREES , OUT) R.move_to(region,IN+RIGHT) - self.play(Write(region)) + self.play(ShowCreation(region)) self.add(R) def get_surface(self,axes, func, **kwargs): @@ -279,8 +280,11 @@ class SurfacesAnimation(ThreeDScene): axes.z_axis.label = z_label for axis in axes: axis.add(axis.label) - return axes + return axes #uploaded by Somnath Pandit.FSF2020_Fubini's_Theorem + + + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3.o_iteration_methods_checkpoint.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3.o_iteration_methods_checkpoint.gif Binary files differnew file mode 100644 index 0000000..2e507f9 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3.o_iteration_methods_checkpoint.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3.o_iteration_methods_checkpoint.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3.o_iteration_methods_checkpoint.py new file mode 100644 index 0000000..55f91d3 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3.o_iteration_methods_checkpoint.py @@ -0,0 +1,226 @@ +from manimlib.imports import * + +class IterationMethods(GraphScene): + CONFIG = { + "x_min" : 0, + "x_max" : 1, + "y_min" : 0, + "y_max" : 1, + "x_tick_frequency" : 1, + "y_tick_frequency" : 1, + "x_labeled_nums": list(np.arange(0,2)), + "y_labeled_nums": list(np.arange(0 ,2)), + "x_axis_width": 6, + "y_axis_height": 6, + "graph_origin": ORIGIN+4*LEFT+3*DOWN, + "area_color": PINK , + "area_opacity": .6, + } + + 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.intro_scene() + self.setup_axes(animate=True) + + + curve1= self.get_graph( + lambda x : x**2 , + x_min = 0, + x_max = 1, + color = ORANGE) + c1_eqn=self.get_graph_label( + curve1, + label="y=x^2", + x_val=.5, + direction=RIGHT, + buff=MED_LARGE_BUFF, + color=ORANGE, + ) + + curve2= self.get_graph( + lambda x : x , + x_min = 0, + x_max = 1, + color = YELLOW) + c2_eqn=self.get_graph_label( + curve2, + label="y=x", + x_val=.5, + direction=LEFT, + buff=MED_LARGE_BUFF, + color=YELLOW, + ) + self.curve1=curve1 + self.curve2=curve2 + + caption_y_int=TextMobject(r"Observe the limits\\ of integration").to_corner(UR) + int_lim=TextMobject( + "$$\\int_0^1$$" + ).next_to( + caption_y_int,DOWN,buff=.5 + ).align_to( + caption_y_int,LEFT + ) + + self.play(ShowCreation(VGroup(curve1,curve2)),Write(VGroup(c2_eqn,c1_eqn))) + rects=self.get_rects() + + self.play(Write(caption_y_int)) + self.show_integral_values_at_different_x() + self.wait(1) + self.add(int_lim) + self.play(FadeOut(self.brace_group)) + self.play(ApplyMethod( + self.y_int.next_to, + int_lim,RIGHT,buff=0)) + + self.play(ApplyMethod( + self.dx_label.next_to, + self.y_int,RIGHT)) + + self.show_area() + + self.wait(2) + + ################### + def intro_scene(self): + text=TextMobject(r"How different orders of \textbf{iterated integral}\\ works over the same region ?" ) + self.play(Write(text),run_time=4) + self.wait(2) + self.play(FadeOut(text)) + + + def show_area(self): + area = self.bounded_riemann_rectangles( + self.curve1, + self.curve2, + x_min = 0, + x_max = 1, + dx =.001, + start_color = self.area_color, + end_color = self.area_color, + fill_opacity = 1, + stroke_width = 0, + ) + self.play(ShowCreation(area)) + # self.transform_between_riemann_rects(self.rects,area) + self.area = area + + def get_rects(self): + rects = self.bounded_riemann_rectangles( + self.curve1, + self.curve2, + x_min = 0, + x_max = 1, + dx =.01, + start_color = self.area_color, + end_color = self.area_color, + fill_opacity =self.area_opacity, + stroke_width = 0, + ) + # self.transform_between_riemann_rects(self.area,rects) + self.rects=rects + return rects + + def show_integral_values_at_different_x(self): + rects=self.rects + rect = rects[len(rects)*1//10] + dx_brace = Brace(rect, DOWN, buff = 0) + dx_label = dx_brace.get_text("$dx$", buff = SMALL_BUFF) + dx_brace_group = VGroup(dx_brace,dx_label) + rp=int(len(rects)/10) + rects_subset = self.rects[4*rp:5*rp] + + last_rect = None + for rect in rects_subset: + brace = Brace(rect, LEFT, buff =.1) + y_int = TexMobject("\\int_{x^2}^{x}dy")#.rotate(PI/2) + y_int.next_to(brace, LEFT, MED_SMALL_BUFF) + anims = [ + rect.set_fill, self.area_color, 1, + dx_brace_group.next_to, rect, DOWN, SMALL_BUFF + ] + if last_rect is not None: + anims += [ + last_rect.set_fill, None, 0, + # last_rect.set_fill, self.area_color, self.area_opacity, + ReplacementTransform(last_brace, brace), + ReplacementTransform(last_y_int, y_int), + ] + else: + anims += [ + GrowFromCenter(brace), + Write(y_int) + ] + self.play(*anims) + # self.wait(.2) + + last_rect = rect + last_brace = brace + last_y_int = y_int + + y_int = last_y_int + y_brace = last_brace + self.brace_group=VGroup(y_brace,dx_brace,rect) + self.y_int=y_int + self.dx_label=dx_label + + + def bounded_riemann_rectangles( + self, + graph1, + graph2, + x_min=None, + x_max=None, + dx=0.01, + input_sample_type="center", + stroke_width=1, + stroke_color=BLACK, + fill_opacity=1, + start_color=None, + end_color=None, + show_signed_area=True, + width_scale_factor=1.001 + ): + x_min = x_min if x_min is not None else self.x_min + x_max = x_max if x_max is not None else self.x_max + if start_color is None: + start_color = self.default_riemann_start_color + if end_color is None: + end_color = self.default_riemann_end_color + rectangles = VGroup() + x_range = np.arange(x_min, x_max, dx) + colors = color_gradient([start_color, end_color], len(x_range)) + for x, color in zip(x_range, colors): + if input_sample_type == "left": + sample_input = x + elif input_sample_type == "right": + sample_input = x + dx + elif input_sample_type == "center": + sample_input = x + 0.5 * dx + else: + raise Exception("Invalid input sample type") + graph1_point = self.input_to_graph_point(sample_input, graph1) + graph1_point_dx= self.input_to_graph_point(sample_input + width_scale_factor * dx, graph1) + graph2_point = self.input_to_graph_point(sample_input, graph2) + + points = VGroup(*list(map(VectorizedPoint, [ + graph1_point, + graph1_point_dx, + graph2_point + ]))) + + rect = Rectangle() + rect.replace(points, stretch=True) + if graph1_point[1] < self.graph_origin[1] and show_signed_area: + fill_color = invert_color(color) + else: + fill_color = color + rect.set_fill(fill_color, opacity=fill_opacity) + rect.set_stroke(stroke_color, width=stroke_width) + rectangles.add(rect) + return rectangles + +#uploaded by Somnath Pandit.FSF2020_Fubini's_Theorem diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3_iteration_methods.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3_iteration_methods.gif Binary files differnew file mode 100644 index 0000000..4e1611b --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3_iteration_methods.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3_iteration_methods.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3_iteration_methods.py new file mode 100644 index 0000000..ad78a0b --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file3_iteration_methods.py @@ -0,0 +1,429 @@ +from manimlib.imports import * + +class IterationMethods(GraphScene): + CONFIG = { + "x_min" : 0, + "x_max" : 1, + "y_min" : 0, + "y_max" : 1, + "x_tick_frequency" : 1, + "y_tick_frequency" : 1, + "x_labeled_nums": list(np.arange(0,2)), + "y_labeled_nums": list(np.arange(0 ,2)), + "x_axis_width": 6, + "y_axis_height": 6, + "graph_origin": ORIGIN+4.5*LEFT+3*DOWN, + "area_color": PINK , + "area_opacity": .6, + } + + 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.intro_scene() + self.setup_axes(animate=True) + + + curve1= self.get_graph( + lambda x : x**2 , + x_min = 0, + x_max = 1, + color = ORANGE) + c1_eqn=self.get_graph_label( + curve1, + label="y=x^2", + x_val=.5, + direction=RIGHT, + buff=MED_LARGE_BUFF, + color=ORANGE, + ) + + curve2= self.get_graph( + lambda x : x , + x_min = 0, + x_max = 1, + color = YELLOW) + c2_eqn=self.get_graph_label( + curve2, + label="y=x", + x_val=.7, + direction=LEFT, + buff=MED_LARGE_BUFF, + color=YELLOW, + ) + self.curve1=curve1 + self.curve2=curve2 + + caption_limit=TextMobject(r"Observe the limits\\ of integration").to_corner(UR) + int_lim=TextMobject( + "$$\\int_0^1$$" + ).next_to( + caption_limit,DOWN,buff=.5 + ).align_to( + caption_limit,LEFT + ) + self.int_lim=int_lim + self.play(ShowCreation(VGroup(curve1,curve2)),Write(VGroup(c2_eqn,c1_eqn))) + + self.play(Write(caption_limit)) + self.get_rects() + self.show_integral_values_at_different_x() + self.wait(1) + self.integral_setup(int_lim,first_y=True) + + + self.another_method_scene() + self.remove(self.area) + self.wait() + + c1_eqn_y=self.get_graph_label( + curve1, + label="x=\sqrt y", + x_val=.6, + direction=RIGHT, + buff=MED_LARGE_BUFF, + color=ORANGE, + ) + c2_eqn_y=self.get_graph_label( + curve2, + label="x=y", + x_val=.7, + direction=LEFT, + buff=MED_LARGE_BUFF, + color=YELLOW, + ) + self.play( + ReplacementTransform(c1_eqn,c1_eqn_y), + ReplacementTransform(c2_eqn,c2_eqn_y) + ) + self.get_rects(base_y=True) + self.show_integral_values_at_different_y() + self.wait(1) + + int_lim_y=int_lim.copy() + int_lim_y.next_to(int_lim,DOWN) + self.int_lim_y=int_lim_y + equal=TextMobject("$$=$$").next_to(int_lim_y,LEFT) + self.add(equal) + + self.integral_setup(int_lim_y,first_y=False) + + self.wait(2) + + ################### + def intro_scene(self): + text=TextMobject(r"How different orders of \textbf{iterated integral}\\ works over the same region ?" ) + self.play(Write(text),run_time=4) + self.wait(2) + self.play(FadeOut(text)) + + def another_method_scene(self): + text=TextMobject(r"The other method\\ of iteration") + text.next_to(self.curve1,UP,buff=-1) + self.play(GrowFromCenter(text)) + self.wait(2) + self.play(LaggedStart(FadeOut(text),lag_ratio=2)) + + def integral_setup(self,ref_object,first_y=True): + if first_y: + area=self.get_area() + self.area=area + self.play(FadeOut(self.brace_group)) + self.play(ApplyMethod( + self.y_int.next_to, + ref_object,RIGHT,buff=0) + ) + + self.play(ApplyMethod( + self.dx_label.next_to, + self.y_int,RIGHT), + ShowCreation(area), + Write(self.int_lim),run_time=4 + ) + else: + area=self.get_area(base_y=True) + self.area=area + self.play( + FadeOut(self.y_brace_group), + Rotate(self.x_int,PI/2) + ) + self.play(ApplyMethod( + self.x_int.next_to, + ref_object,RIGHT,buff=0) + ) + self.play(ApplyMethod( + self.dy_label.next_to, + self.x_int,RIGHT), + ShowCreation(area), + Write(self.int_lim_y),run_time=4 + ) + + def get_area(self,base_y=False): + if base_y: + area = self.bounded_riemann_rectangles_y( + lambda x: x, + lambda x: np.sqrt(x), + y_min = 0, + y_max = 1, + dy =.001, + start_color = self.area_color, + end_color = self.area_color, + fill_opacity =self.area_opacity, + stroke_width = 0, + ) + self.y_area = area + else: + area = self.bounded_riemann_rectangles( + self.curve1, + self.curve2, + x_min = 0, + x_max = 1, + dx =.001, + start_color = self.area_color, + end_color = self.area_color, + fill_opacity =self.area_opacity, + stroke_width = 0, + ) + self.area = area + + # self.transform_between_riemann_rects(self.rects,area) + return area + + def get_rects(self,base_y=False): + if base_y: + rects = self.bounded_riemann_rectangles_y( + lambda x: x, + lambda x: np.sqrt(x), + y_min = 0, + y_max = 1, + dy =.01, + start_color = self.area_color, + end_color = self.area_color, + fill_opacity =self.area_opacity, + stroke_width = 0, + ) + self.y_rects=rects + else: + rects = self.bounded_riemann_rectangles( + self.curve1, + self.curve2, + x_min = 0, + x_max = 1, + dx =.01, + start_color = self.area_color, + end_color = self.area_color, + fill_opacity =self.area_opacity, + stroke_width = 0, + ) + self.rects=rects + # self.transform_between_riemann_rects(self.area,rects) + + return rects + + def show_integral_values_at_different_x(self): + rects=self.rects + rect = rects[len(rects)*1//10] + dx_brace = Brace(rect, DOWN, buff = 0) + dx_label = dx_brace.get_text("$dx$", buff = SMALL_BUFF) + dx_brace_group = VGroup(dx_brace,dx_label) + rp=int(len(rects)/20) + rects_subset = rects[6*rp:7*rp] + + last_rect = None + for rect in rects_subset: + brace = Brace(rect, LEFT, buff =.1) + y_int = TexMobject("\\int_{x^2}^{x}dy")#.rotate(PI/2) + y_int.next_to(brace, LEFT, MED_SMALL_BUFF) + anims = [ + rect.set_fill, self.area_color, 1, + dx_brace_group.next_to, rect, DOWN, SMALL_BUFF + ] + if last_rect is not None: + anims += [ + last_rect.set_fill, None, 0, + # last_rect.set_fill, self.area_color, self.area_opacity, + ReplacementTransform(last_brace, brace), + ReplacementTransform(last_y_int, y_int), + ] + else: + anims += [ + GrowFromCenter(brace), + Write(y_int) + ] + self.play(*anims) + # self.wait(.2) + + last_rect = rect + last_brace = brace + last_y_int = y_int + + y_int = last_y_int + y_brace = last_brace + self.brace_group=VGroup(y_brace,dx_brace,rect) + self.y_int=y_int + self.dx_label=dx_label + + def show_integral_values_at_different_y(self): + rects=self.y_rects + rect = rects[len(rects)*1//10] + dy_brace = Brace(rect, LEFT, buff = 0) + dy_label = dy_brace.get_text("$dy$", buff = SMALL_BUFF) + dy_brace_group = VGroup(dy_brace,dy_label) + rp=int(len(rects)/20) + rects_subset = rects[5*rp:6*rp] + + last_rect = None + for rect in rects_subset: + brace = Brace(rect, DOWN, buff =.1) + x_int = TexMobject("\\int_{y}^{\sqrt y}dx").rotate(-PI/2) + x_int.next_to(brace, DOWN, SMALL_BUFF) + anims = [ + rect.set_fill, self.area_color, 1, + dy_brace_group.next_to, rect, LEFT, SMALL_BUFF + ] + if last_rect is not None: + anims += [ + last_rect.set_fill, None, 0, + # last_rect.set_fill, self.area_color, self.area_opacity, + ReplacementTransform(last_brace, brace), + ReplacementTransform(last_x_int, x_int), + ] + else: + anims += [ + GrowFromCenter(brace), + Write(x_int) + ] + self.play(*anims) + # self.wait(.2) + + last_rect = rect + last_brace = brace + last_x_int = x_int + + x_int = last_x_int + y_brace = last_brace + self.y_brace_group=VGroup(y_brace,dy_brace,rect) + self.x_int=x_int + self.dy_label=dy_label + + + def bounded_riemann_rectangles( + self, + graph1, + graph2, + x_min=None, + x_max=None, + dx=0.01, + input_sample_type="center", + stroke_width=1, + stroke_color=BLACK, + fill_opacity=1, + start_color=None, + end_color=None, + show_signed_area=True, + width_scale_factor=1.001 + ): + x_min = x_min if x_min is not None else self.x_min + x_max = x_max if x_max is not None else self.x_max + if start_color is None: + start_color = self.default_riemann_start_color + if end_color is None: + end_color = self.default_riemann_end_color + rectangles = VGroup() + x_range = np.arange(x_min, x_max, dx) + colors = color_gradient([start_color, end_color], len(x_range)) + for x, color in zip(x_range, colors): + if input_sample_type == "left": + sample_input = x + elif input_sample_type == "right": + sample_input = x + dx + elif input_sample_type == "center": + sample_input = x + 0.5 * dx + else: + raise Exception("Invalid input sample type") + graph1_point = self.input_to_graph_point(sample_input, graph1) + graph1_point_dx= self.input_to_graph_point(sample_input + width_scale_factor * dx, graph1) + graph2_point = self.input_to_graph_point(sample_input, graph2) + + points = VGroup(*list(map(VectorizedPoint, [ + graph1_point, + graph1_point_dx, + graph2_point + ]))) + + rect = Rectangle() + rect.replace(points, stretch=True) + if graph1_point[1] < self.graph_origin[1] and show_signed_area: + fill_color = invert_color(color) + else: + fill_color = color + rect.set_fill(fill_color, opacity=fill_opacity) + rect.set_stroke(stroke_color, width=stroke_width) + rectangles.add(rect) + return rectangles + + def bounded_riemann_rectangles_y( + self, + graph1, + graph2, + y_min=None, + y_max=None, + dy=0.01, + input_sample_type="center", + stroke_width=1, + stroke_color=BLACK, + fill_opacity=1, + start_color=None, + end_color=None, + show_signed_area=True, + width_scale_factor=1.001 + ): + y_min = y_min if y_min is not None else self.y_min + y_max = y_max if y_max is not None else self.y_max + if start_color is None: + start_color = self.default_riemann_start_color + if end_color is None: + end_color = self.default_riemann_end_color + rectangles = VGroup() + y_range = np.arange(y_min, y_max, dy) + colors = color_gradient([start_color, end_color], len(y_range)) + for y, color in zip(y_range, colors): + if input_sample_type == "left": + sample_input = y + elif input_sample_type == "right": + sample_input = y + dy + elif input_sample_type == "center": + sample_input = y + 0.5 * dy + else: + raise Exception("Invalid input sample type") + graph1_point = self.coords_to_point( + graph1(sample_input),sample_input + ) + dy_input=sample_input + width_scale_factor * dy + graph1_point_dy= self.coords_to_point( + graph1(dy_input),dy_input + ) + graph2_point = self.coords_to_point( + graph2(sample_input),sample_input + ) + + points = VGroup(*list(map(VectorizedPoint, [ + graph1_point, + graph1_point_dy, + graph2_point + ]))) + + rect = Rectangle() + rect.replace(points, stretch=True) + if graph1_point[1] < self.graph_origin[1] and show_signed_area: + fill_color = invert_color(color) + else: + fill_color = color + rect.set_fill(fill_color, opacity=fill_opacity) + rect.set_stroke(stroke_color, width=stroke_width) + rectangles.add(rect) + return rectangles + + +#uploaded by Somnath Pandit.FSF2020_Fubini's_Theorem diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file4_curvy_region.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file4_curvy_region.gif Binary files differnew file mode 100644 index 0000000..b0620e5 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file4_curvy_region.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file4_curvy_region.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file4_curvy_region.py new file mode 100644 index 0000000..46134a7 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's-theorem/file4_curvy_region.py @@ -0,0 +1,102 @@ +from manimlib.imports import * + +class CurvyRegion(GraphScene): + CONFIG = { + "x_min": 0, + "x_max": 8, + "y_min": 0, + "y_max": 6, + "graph_origin": ORIGIN+4.5*LEFT+3*DOWN, + "x_labeled_nums": np.arange(0, 9,2), + "y_labeled_nums": np.arange(0, 7,2), + "x_axis_width": 6, + "y_axis_height": 6, + } + + def construct(self): + XD = self.x_axis_width/(self.x_max- self.x_min) + YD = self.y_axis_height/(self.y_max- self.y_min) + self.X=XD*RIGHT ;self.Y=YD*UP + + sin_curve_points=[self.graph_origin+(2+.5*np.sin(2*y),y,0) + for y in np.arange(1,5,.005)] + + cos_curve_points=[self.graph_origin+( + 5+.5*np.cos(2*y),y,0) + for y in np.arange(1,5,.005)] + cos_curve_points.reverse() + + region=Polygon( + *sin_curve_points+cos_curve_points, + color=YELLOW, + stroke_width=1, + fill_color=BLUE_E, + fill_opacity=.75 + ) + + line=Line((1,0,0),(1,6,0),color=RED) + line.move_to(self.graph_origin+2.5*self.X,DOWN) + self.line=line + self.setup_axes(animate = False) + + self.add(region) + self.wait() + self.first_y_int_scene() + self.try_x_first_scene() + + + def first_y_int_scene(self): + talk=TextMobject(r"For doing the $y$ integration\\ first we need to set\\ proper $y$ limts").to_corner(UR,buff=LARGE_BUFF) + problem=TextMobject(r"But here we get\\ more than two $y$ values\\ for a single $x$ value" ).to_corner(UR,buff=LARGE_BUFF) + int_y=TextMobject("$$\\int_?^? dy$$").next_to(problem,DOWN,buff=.5) + + self.play(Write(talk)) + self.play(FadeIn(self.line)) + self.wait(2) + self.play(ReplacementTransform(talk,problem)) + self.play( + ApplyMethod(self.line.shift,3.7*self.X), + run_time=4 + ) + self.wait() + self.play(Write(int_y)) + self.wait(3) + self.play(FadeOut(VGroup(problem,int_y,self.line))) + + def try_x_first_scene(self): + try_text=TextMobject(r"But if we try to integrate\\ along $x$ first ...." ).to_corner(UR,buff=LARGE_BUFF) + good_limits=TextMobject(r"For one $y$ value we get\\ only \textbf{two} $x$ values $\dots$").to_corner(UR,buff=LARGE_BUFF) + limit_values= TextMobject(r"one Lower limit\\ one Upper limit ").next_to(good_limits,DOWN,buff=.5) + int_x=TextMobject("$$\\int_{f(y)}^{g(y)} dx$$").next_to(limit_values,DOWN) + + self.setup_line() + self.play(Write(try_text)) + self.play(FadeIn(self.line)) + self.wait() + self.play(ReplacementTransform(try_text,good_limits)) + self.wait() + self.play( + ApplyMethod(self.line.shift,3*self.Y), + run_time=4 + ) + self.play(Write(limit_values)) + self.wait() + self.show_functions() + self.play(Write(int_x)) + self.wait(3) + + def setup_line(self): + line=self.line.rotate(PI/2) + line.move_to(self.graph_origin+.5*self.X+1.5*self.Y,LEFT) + self.line=line + + def show_functions(self): + fy=TextMobject("$$f(y)$$") + gy=TextMobject("$$g(y)$$") + fy.move_to(self.graph_origin+2*self.X+3.3*self.Y) + gy.move_to(self.graph_origin+7*self.X+2*self.Y) + self.play(FadeIn(VGroup(fy,gy))) + + + #uploaded by Somnath Pandit.FSF2020_Fubini's_Theorem + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's_theorem/surface2.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's_theorem/surface2.gif Binary files differdeleted file mode 100644 index ac13f21..0000000 --- a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fubini's_theorem/surface2.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/README.md b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/README.md new file mode 100644 index 0000000..aa8c7f8 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/README.md @@ -0,0 +1,9 @@ +**file1_grad_of_scalar_function** + + +**file2_line_int_independent_of_path** + + +**file3_line_int_example** + + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file1_grad_of_scalar_function.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file1_grad_of_scalar_function.gif Binary files differnew file mode 100644 index 0000000..5a6e102 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file1_grad_of_scalar_function.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file1_grad_of_scalar_function.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file1_grad_of_scalar_function.py new file mode 100644 index 0000000..c9f479c --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file1_grad_of_scalar_function.py @@ -0,0 +1,308 @@ +from manimlib.imports import * + +class GradOfScalar(ThreeDScene): + + CONFIG = { + "axes_config": { + "x_min": -3, + "x_max": 3, + "y_min": -3, + "y_max": 3, + "z_min": 0, + "z_max": 3, + "a":-3 ,"b": 3, "c":-3 , "d":3, + "axes_shift": ORIGIN+IN, + "x_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "y_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "z_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "num_axis_pieces": 1, + }, + "default_graph_style": { + "stroke_width": 2, + "stroke_color": WHITE, + }, + "default_vector_field_config": { + "delta_x": 1, + "delta_y": 1, + "x_min": -3, + "x_max": 3, + "y_min": -3, + "y_max": 3, + "min_magnitude": 0, + "max_magnitude": 2, + "colors": [TEAL,GREEN,GREEN,GREEN,YELLOW,RED], + "length_func": lambda norm : norm*np.exp(-.38*norm)/2, + "opacity": 1.0, + "vector_config": { + "stroke_width":8 + }, + }, + "default_surface_config": { + "fill_opacity": 0.5, + "checkerboard_colors": [BLUE_E], + "stroke_width": .5, + "stroke_color": WHITE, + "stroke_opacity": 0.75, + }, + } + + + def construct(self): + + self.setup_axes() + axes=self.axes + + self.set_camera_orientation(distance=35, + phi=70 * DEGREES, + theta=-135 * DEGREES, + ) + + scalar_fn_text=TexMobject("f(x,y,z)=","xy").set_color(BLUE) + scalar_fn_text.to_corner(UR,buff=.6) + + operator=TexMobject("\\vec\\nabla").next_to( + scalar_fn_text,LEFT,buff=.2 + ).set_color(GOLD) + grad_text=TexMobject(r"\dfrac{\partial f}{\partial x} \hat i+\dfrac{\partial f}{\partial y} \hat j+\dfrac{\partial f}{\partial z} \hat k").set_color(GOLD) + grad_text.next_to(scalar_fn_text,DOWN).scale(.9) + + VGroup(grad_text[0][1],grad_text[0][9],grad_text[0][17]).set_color(BLUE) + VGroup(grad_text[0][5:8],grad_text[0][13:16],grad_text[0][21:23]).set_color(WHITE) + + vector_field_text=TexMobject("\\vec F=y\hat i+x\hat j").set_color_by_gradient(*self.default_vector_field_config["colors"]) + vector_field_text.next_to(scalar_fn_text,DOWN) + + + #always generate the scalar field first + s_field1=self.get_scalar_field( + func= lambda u ,v : u*v/7 + ) + v_field1=self.get_vector_field( + lambda v: np.array([ + v[1], + v[0], + 0, + ]), + on_surface=True, + ) + + self.add_fixed_in_frame_mobjects(scalar_fn_text) + + self.begin_ambient_camera_rotation(rate=.2) + self.play(Write(s_field1)) + self.wait(1) + self.stop_ambient_camera_rotation() + + self.add_fixed_in_frame_mobjects(operator) + self.play(Write(operator),FadeOut(scalar_fn_text[1])) + self.add_fixed_in_frame_mobjects(grad_text) + self.play(Write(grad_text)) + self.wait(2) + + self.play(FadeOut(grad_text)) + self.add_fixed_in_frame_mobjects(vector_field_text) + show_vec_field=[ + FadeIn(v_field1), + Write(vector_field_text), + ] + + self.begin_ambient_camera_rotation(rate=.2) + self.move_camera( + # distance=20, + phi=60 * DEGREES, + added_anims=show_vec_field, + run_time=4.5 + ) + + self.wait(2) + self.stop_ambient_camera_rotation() + + fadeout= [FadeOut(s_field1)] + self.move_camera( + # distance=20, + phi=0 * DEGREES, + theta=-90 * DEGREES, + added_anims=fadeout, + run_time=2 + ) + self.wait(2) + + + + + + def get_scalar_field(self,func,**kwargs): + surface= self.get_surface( + lambda x , y: + func(x,y), + ) + + self.surface_points=self.get_points(func) + return surface + + def get_points(self,func): + axes=self.axes + dn=.5 + x_vals=np.arange(axes.a,axes.b,dn) + y_vals=np.arange(axes.c,axes.d,dn) + points=[] + for x_val in x_vals: + for y_val in y_vals: + points+=[axes.c2p(x_val,y_val,func(x_val,y_val)+.05)] + return points + + def get_vector_field(self,func,on_surface=True,**kwargs): + config = dict() + config.update(self.default_vector_field_config) + config.update(kwargs) + vector_field= VectorField(func,**config) + self.vector_field=vector_field + + if on_surface: + vector_field=self.get_vectors_on_surface() + + return vector_field + + + + def get_vectors_on_surface(self): + config = dict() + config.update(self.default_vector_field_config["vector_config"]) + vectors_on_surface = VGroup(*[ + self.vector_field.get_vector(point,**config) + for point in self.surface_points + ]) + + return vectors_on_surface + + + def get_surface(self, func, **kwargs): + axes=self.axes + config = { + "u_min": axes.a, + "u_max": axes.b, + "v_min": axes.c, + "v_max": axes.d, + "resolution": ( + 2*(axes.y_max - axes.y_min) // axes.y_axis.tick_frequency, + (axes.x_max - axes.x_min) // axes.x_axis.tick_frequency, + ), + } + + config.update(self.default_surface_config) + config.update(kwargs) + return ParametricSurface( + lambda x,y : axes.c2p( + x, y, func(x, y) + ), + **config + ) + + + +#------------------------------------------------------- + #customize 3D axes + def get_three_d_axes(self, include_labels=True, include_numbers=False, **kwargs): + config = dict(self.axes_config) + config.update(kwargs) + axes = ThreeDAxes(**config) + axes.set_stroke(width=2) + self.axes=axes + + if include_numbers: + self.add_axes_numbers(axes) + + if include_labels: + self.add_axes_labels(axes) + + # Adjust axis orientation + axes.x_axis.rotate( + -90 * DEGREES, LEFT, + about_point=axes.c2p(0, 0, 0), + ) + axes.y_axis.rotate( + 90 * DEGREES, UP, + about_point=axes.c2p(0, 0, 0), + ) + + return axes + + + def setup_axes(self): + axes = self.get_three_d_axes(include_labels=True) + axes.scale(1) + # axes.center() + axes.shift(axes.axes_shift) + + self.add(axes) + self.axes = axes + + def add_axes_numbers(self, axes): + x_axis = axes.x_axis + y_axis = axes.y_axis + tex_vals_x = [ + + ("1", axes.b), + ("-1", axes.a), + ] + tex_vals_y=[ + + ("1", axes.d) + ] + x_labels = VGroup() + y_labels = VGroup() + for tex, val in tex_vals_x: + label = TexMobject(tex) + label.scale(1) + label.next_to(x_axis.n2p(val), DOWN) + # label.rotate(180 * DEGREES) + x_labels.add(label) + x_axis.add(x_labels) + x_axis.numbers = x_labels + + for tex, val in tex_vals_y: + label = TexMobject(tex) + label.scale(1) + label.next_to(y_axis.n2p(val), LEFT) + label.rotate(90 * DEGREES) + y_labels.add(label) + + y_axis.add(y_labels) + y_axis.numbers = y_labels + + return axes + + def add_axes_labels(self, axes): + x_label = TexMobject("x") + x_label.next_to(axes.x_axis.get_end(), RIGHT) + axes.x_axis.label = x_label + + y_label = TextMobject("y") + y_label.rotate(90 * DEGREES, OUT) + y_label.next_to(axes.y_axis.get_end(), UP) + axes.y_axis.label = y_label + + z_label = TextMobject("z") + z_label.rotate(90 * DEGREES, RIGHT) + z_label.next_to(axes.z_axis.get_zenith(), LEFT) + axes.z_axis.label = z_label + for axis in axes: + axis.add(axis.label) + return axes + + + + #uploaded by Somnath Pandit. FSF2020_Fundamental_Theorem_of_Line_Integrals + + + + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file2_line_int_independent_of_path.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file2_line_int_independent_of_path.gif Binary files differnew file mode 100644 index 0000000..29c6d02 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file2_line_int_independent_of_path.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file2_line_int_independent_of_path.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file2_line_int_independent_of_path.py new file mode 100644 index 0000000..b9597b6 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file2_line_int_independent_of_path.py @@ -0,0 +1,174 @@ +from manimlib.imports import * + + +class LineIntegration(GraphScene): + CONFIG = { + "x_min" : -5, + "x_max" : 5, + "y_min" : -5, + "y_max" : 5, + "axes_color":BLACK, + "graph_origin": ORIGIN+1.2*DOWN, + "x_axis_width": 10, + "y_axis_height": 10 , + "x_axis_label": "", + "y_axis_label": "", + "x_tick_frequency": 1, + "y_tick_frequency": 1, + "default_vector_field_config": { + "delta_x": .6, + "delta_y": .6, + "min_magnitude": 0, + "max_magnitude": .5, + "colors": [GREEN,BLUE,BLUE,TEAL], + "length_func": lambda norm : .45*sigmoid(norm), + "opacity": .75, + "vector_config": { + "stroke_width":1.5 + }, + }, + + "a": .45,"b": 2, + "path_color": PURPLE + } + + 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) + + + + + vector_field=self.get_vector_field( + lambda v: np.array([ + v[1]-self.graph_origin[1], + v[0]-self.graph_origin[0], + 0, + ]) + ) + vector_field_text=TexMobject( + "\\vec F(x,y)","=y\hat i+x\hat j", + stroke_width=1.5 + ).to_edge(TOP,buff=.2) + + vector_field_text[0][0:2].set_color(TEAL) + + grad_f=TexMobject( + "\\vec\\nabla f(x,y)", + stroke_width=1.5 + ) + grad_f[0][2].set_color(LIGHT_BROWN) + grad_f.move_to(vector_field_text[0]) + + self.add(vector_field,) + self.play(Write(vector_field_text)) + self.wait() + self.play( + ReplacementTransform( + vector_field_text[0],grad_f + ) + ) + self.get_endpoints_of_curve() + self.wait(.6) + vector_field.set_fill(opacity=.4) + self.show_line_integral() + 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_endpoints_of_curve(self): + points=[[-3,0],[2,2]] + point_labels= ["P_i","P_f"] + for point,label in zip(points,point_labels): + dot=Dot(self.coords_to_point(*point)).set_color(RED) + dot_label=TexMobject(label) + dot_label.next_to(dot,DR,buff=.2) + self.play(FadeIn(VGroup(dot,dot_label))) + self.wait(.2) + + self.end_points=points + + def show_line_integral(self): + int_text=TexMobject( + r"\int_{P_i}^{P_f}\vec F \cdot d\vec r", + stroke_width=1.5, + ).scale(1.2) + int_text[0][0].set_color(self.path_color) + int_text[0][5:7].set_color(TEAL) + int_text.to_edge(RIGHT+UP,buff=1) + + int_value= TexMobject(r"=f(P_i)-f(P_f)", + stroke_width=1.5 + ).next_to(int_text,DOWN) + VGroup(int_value[0][1], + int_value[0][7] + ).set_color(LIGHT_BROWN) + + path_indepent_text=TextMobject( + r"Value of the Line Integral is\\ independent of Path",color=GOLD,stroke_width=2,).to_corner(DR,buff=1) + + path_indepent_text[0][-4:].set_color(self.path_color) + + + self.play(Write(VGroup( + int_text,int_value + )), + run_time=2 + ) + self.wait(1.5) + + + self.show_path([[0,1],[-1,2],[1,3]]) + self.play(Indicate(int_value)) + self.play(Uncreate(self.path)) + + self.show_path([[0,1]]) + self.play(Indicate(int_value)) + self.play(Uncreate(self.path)) + + self.show_path([[-1,1],[-1,-2],[-5,0],[-2,3.5],[1,1]]) + self.play(Indicate(int_value),run_time=2) + self.wait(.6) + + self.play(Write(path_indepent_text)) + + + + def show_path(self,points): + points=[self.end_points[0]]+points+[self.end_points[1]] + + path= VMobject() + path.set_points_smoothly([ + self.coords_to_point(*point) + for point in points + ]) + path.set_color(self.path_color) + self.play(ShowCreation(path),run_time=1.5) + + self.path=path + + + + + +#uploaded by Somnath Pandit. FSF2020_Fundamental_Theorem_of_Line_Integrals + + + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file3_line_int_example.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file3_line_int_example.gif Binary files differnew file mode 100644 index 0000000..20ed081 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file3_line_int_example.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file3_line_int_example.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file3_line_int_example.py new file mode 100644 index 0000000..71506a3 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/file3_line_int_example.py @@ -0,0 +1,149 @@ +from manimlib.imports import * + + +class LineIntegration(GraphScene): + CONFIG = { + "x_min" : -1, + "x_max" : 2, + "y_min" : -1, + "y_max" : 2, + "graph_origin": ORIGIN+3*LEFT+1.5*DOWN, + "x_axis_width": 10, + "y_axis_height": 10 , + "x_tick_frequency": 1, + "y_tick_frequency": 1, + "default_vector_field_config": { + "delta_x": .5, + "delta_y": .5, + "min_magnitude": 0, + "max_magnitude": .5, + "colors": [GREEN,BLUE,BLUE,TEAL], + "length_func": lambda norm : .4*sigmoid(norm), + "opacity": .75, + "vector_config": { + "stroke_width":2 + }, + }, + + "a": .45,"b": 2, + } + + 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) + + + + + vector_field=self.get_vector_field( + lambda v: np.array([ + v[1]-self.graph_origin[1], + v[0]-self.graph_origin[0], + 0, + ]) + ) + vector_field_text=TexMobject( + "\\vec F=y\hat i+x\hat j", + stroke_width=2 + ).to_corner(UR,buff=.75).scale(1.2) + + vector_field_text[0][0:3].set_color(TEAL), + self.add(vector_field,) + self.play(Write(vector_field_text)) + self.wait() + self.get_endpoints_of_curve() + self.wait(.6) + self.play( + vector_field_text.shift,5*LEFT, + + ) + vector_field.set_fill(opacity=.2) + self.show_line_integral() + 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_endpoints_of_curve(self): + points=[[1,1],[0,0]] + point_labels= ["(1,1)","(0,0)"] + for point,label in zip(points,point_labels): + dot=Dot(self.coords_to_point(*point)).set_color(RED) + dot_label=TexMobject(label) + dot_label.next_to(dot,DR) + self.add(dot,dot_label) + self.end_points=points + + def show_line_integral(self): + int_text=TexMobject( + "\\int_\\text{\\textbf{path}}\\vec F \\cdot d\\vec r= 1", + color=BLUE, + stroke_width=1.5 + ).scale(1.2) + int_text[0][0].set_color(RED_C) + int_text[0][5:7].set_color(TEAL) + int_text.to_edge(RIGHT+UP,buff=1) + + close_int=TexMobject("O").set_color(RED).scale(1.3) + close_int.move_to(int_text[0][0],OUT) + close_int_val=TexMobject("0",color=BLUE).scale(1.4) + close_int_val.move_to(int_text[0][-1],OUT) + + self.play(Write(int_text)) + + + self.show_method([[0,1]]) + self.play(Indicate(int_text)) + self.wait() + + self.show_method([[1,0]]) + self.play(Indicate(int_text)) + self.wait() + self.remove(int_text[0][-1]) + self.add(close_int) + + for i in range(2): + self.play(self.paths[i].rotate,PI) + self.play(Indicate(close_int)) + self.play(Write(close_int_val)) + self.wait() + + + def show_method(self,points): + points=points+self.end_points + paths=[] + for i in range(-1,len(points)-2): + path=Arrow( + self.coords_to_point(*points[i]), + self.coords_to_point(*points[i+1]), + buff=0 + ).set_color(BLUE) + paths+=VGroup(path) + self.play(GrowArrow(path),run_time=1.5) + + self.paths=paths + + + + + +#uploaded by Somnath Pandit. FSF2020_Fundamental_Theorem_of_Line_Integrals + + + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/README.md b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/README.md new file mode 100644 index 0000000..17077b6 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/README.md @@ -0,0 +1,14 @@ +**file1_scalar_line_int_as_sum** + + +**file2_scalar_line_integral** + + + +**file3_vector_line_int_as_sum** + + + + +**file4_helix** + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file1_scalar_line_int_as_sum.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file1_scalar_line_int_as_sum.gif Binary files differnew file mode 100644 index 0000000..1984b08 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file1_scalar_line_int_as_sum.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file1_scalar_line_int_as_sum.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file1_scalar_line_int_as_sum.py new file mode 100644 index 0000000..e3f3574 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file1_scalar_line_int_as_sum.py @@ -0,0 +1,227 @@ +from manimlib.imports import * + + +class LineIntegrationAsSum(GraphScene): + CONFIG = { + "x_min" : 0, + "x_max" : 10, + "y_min" : 0, + "y_max" : 6, + "graph_origin": ORIGIN+5*LEFT+3*DOWN, + "x_axis_width": 10, + "y_axis_height": 6 , + "x_tick_frequency": 2, + "y_tick_frequency": 2, + "Func":lambda x : 1+x**1.3*np.exp(-.12*(x-2)**2)*np.sin(x/4), + "a": 1 ,"b": 9, "n": 15, + } + + 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) + + curve=self.get_graph( + self.Func, + x_min=self.a, + x_max=self.b, + ) + curve.set_color([BLACK,BLUE,BLUE,BLUE,BLACK]) + curve_label= self.get_graph_label( + curve, + label="\\text{path of intgration}", + x_val=4, + direction=UR, + buff=.6, + color=BLUE + ) + self.curve=curve + self.curve_label=curve_label + + self.play(ShowCreation(VGroup(curve,curve_label))) + self.wait(.6) + self.break_in_arcs() + self.show_the_sum() + self.construct_equation() + self.wait(2) + + + + def break_in_arcs(self): + + self.write_about_breaking() + + dl=0.8 + self.get_breakers(dl) + self.wait(2) + self.play(FadeOut(self.upto_break_text)) + self.dl=dl + + def write_about_breaking(self): + breaking_text=TextMobject("\\texttt{..broken}"," into small", "subarcs") + breaking_text.set_color_by_tex_to_color_map({ + "broken":RED,"subarcs": BLUE + }) + breaking_text.next_to(self.curve_label,DOWN) + breaking_text.align_to(self.curve_label,LEFT) + self.play( + Write(breaking_text) + ) + + self.upto_break_text=VGroup( + self.curve_label, + breaking_text, + ) + + def get_breakers(self,dl): + point=self.a + points=[] + while point<(self.b-dl) : + start=point + end=point+dl + points += [end] + breaker=Line( + self.input_to_graph_point(start,self.curve), + self.input_to_graph_point(end,self.curve), + stroke_width=2, + color=RED, + ) + breaker.rotate(PI/2).scale(.5) + + point=end + self.play(FadeIn(breaker),run_time=.2) + # self.add(breaker) + + del points[-1] + self.points=points + + + def show_the_sum(self): + at_any_points_text=TextMobject("At any ","point", "in each ", "subarc") + at_any_points_text.set_color_by_tex_to_color_map({ + "point":YELLOW , "subarc": BLUE + }) + at_any_points_text.to_edge(TOP,buff=SMALL_BUFF) + + evaluate_text=TextMobject("$f(x,y)$ ", "is evaluated").next_to(at_any_points_text,DOWN) + evaluate_text.set_color_by_tex("$f(x,y)$",ORANGE) + + self.at_any_points_text=at_any_points_text + self.evaluate_text=evaluate_text + + + dots=[] + for point in self.points: + + dot=Dot( + point=self.input_to_graph_point(point,self.curve), + radius= .7*DEFAULT_DOT_RADIUS, + stroke_width= 0, + fill_opacity= 1.0, + color= YELLOW, + ) + dots+=[dot] + + self.play( + Write(at_any_points_text), + FadeIn(VGroup(*dots)),run_time=1.5 + ) + self.wait() + self.position_of_point_irrelevent() + self.multiply_with_function(dots) + + + + def multiply_with_function(self,dots): + index=-(len(self.points)//3) + dot=dots[index] + + + multiply_text=TexMobject("f(x_i,y_i)", "\\text{ is multiplied with }","\\Delta s_i") + multiply_text.set_color_by_tex_to_color_map({ + "f(x_i,y_i)":ORANGE , "\\Delta s_i": BLUE + }) + multiply_text.to_edge(TOP,buff=MED_SMALL_BUFF) + + point_coord=TextMobject("$(x_i,y_i)$",color=YELLOW) + point_coord.next_to(dot,DL,buff=.01).scale(.8) + + func_val=TextMobject("$f(x_i,y_i)$",color=ORANGE) + func_val.next_to(dot,UR) + + sum_up_text=TextMobject("and "," summed ", "for all i' s") + sum_up_text.set_color_by_tex("summed",PURPLE) + sum_up_text.next_to(multiply_text,DOWN) + + dot.set_color(ORANGE).scale(1.2) + + self.play(FadeIn(VGroup( + point_coord,dot + ))) + self.play(Write(self.evaluate_text)) + self.play(Write(func_val)) + self.play(FadeIn(VGroup(*[ + dot.set_color(ORANGE).scale(1.4) + for dot in dots ] + ))) + self.wait(2) + self.remove(point_coord) + self.get_ds(dots,index) + self.play(GrowFromCenter(self.ds_brace_group)) + self.wait(2) + self.play(FadeOut(VGroup( + self.ds_brace, + self.at_any_points_text, + self.evaluate_text + ))) + self.play(Write(multiply_text)) + self.play(ApplyMethod( + self.ds_brace_label.next_to, + func_val, RIGHT,buff=.2 + )) + self.play(Write(sum_up_text)) + + self.func_val=func_val + self.sum_text_group=VGroup(multiply_text,sum_up_text) + + def position_of_point_irrelevent(self): + pass + + + + def get_ds(self,dots,index): + p1= dots[index] + p2= dots[index+1] + ds_brace=Brace(VGroup(p1,p2),DL) + ds_brace.move_to(p1,UR) + ds_brace_label=ds_brace.get_text("$\Delta s_i$", buff = .05) + ds_brace_label.set_color(BLUE) + self.ds_brace=ds_brace + self.ds_brace_label=ds_brace_label + self.ds_brace_group=VGroup(ds_brace,ds_brace_label) + + + def construct_equation(self): + sum_eqn=TextMobject("$$\\sum_i^{ } $$").set_color(PURPLE) + sum_eqn.move_to(self.graph_origin+7*self.X+4*self.Y) + + line_integral_text=TextMobject("The Value of the line integral is").next_to(self.sum_text_group,IN) + approx=TextMobject("$\\approx$",color=RED).next_to(sum_eqn,LEFT) + multipled=VGroup(self.func_val,self.ds_brace_label) + self.play(FadeIn(sum_eqn)) + self.play(ApplyMethod( + multipled.next_to,sum_eqn,RIGHT + )) + self.wait() + self.play(FadeOut(self.sum_text_group)) + self.play(Write(line_integral_text)) + self.play(FadeIn(approx)) + + + +#uploaded by Somnath Pandit.FSF2020_Line Integrals + + + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file2_scalar_line_integral.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file2_scalar_line_integral.gif Binary files differnew file mode 100644 index 0000000..71c97d6 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file2_scalar_line_integral.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file2_scalar_line_integral.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file2_scalar_line_integral.py new file mode 100644 index 0000000..996ead1 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file2_scalar_line_integral.py @@ -0,0 +1,421 @@ +from manimlib.imports import * + +class LineIntegrationProcess(SpecialThreeDScene): + + CONFIG = { + "axes_config": { + "x_min": -4, + "x_max": 4, + "y_min": 0, + "y_max": 4, + "z_min": 0, + "z_max": 4, + "a":-3 ,"b": 3, "c":0 , "d":3.5, + "axes_shift":3*IN, + "x_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "y_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "z_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + "num_axis_pieces": 1, + }, + "default_graph_style": { + "stroke_width": 2, + "stroke_color": WHITE, + }, + "default_surface_config": { + "fill_opacity": 0.5, + "checkerboard_colors": [LIGHT_GREY], + "stroke_width": 0.2, + "stroke_color": WHITE, + "stroke_opacity": 0.75, + }, + "Func": lambda x,y: 1+x**2*y/15 + } + + + def construct(self): + + self.setup_axes() + axes=self.axes + + self.set_camera_orientation(distance=35, + phi=65 * DEGREES, + theta=-65 * DEGREES, + ) + + fn_text=TextMobject("$z=2+x^2y$").set_color(BLUE) + fn_text.to_corner(UR,buff=.8).shift(DOWN) + + #get the surface + surface= self.get_surface( + lambda x , y: + self.Func(x,y) + ) + surface.set_style( + fill_opacity=0.5, + fill_color=BLUE_D, + stroke_width=0.5, + stroke_color=WHITE, + ) + + + # self.play(Write(surface)) + self.add_fixed_in_frame_mobjects(fn_text) + self.play(Write(surface),Write(fn_text)) + self.get_line_of_int() + self.begin_ambient_camera_rotation(rate=-0.035) + self.get_field_values_on_line() + self.wait(1.5) + self.area=self.get_area() + area_text=TextMobject("Line"," Integral in the",r" scalar field\\"," means this" ,"area") + area_text.set_color_by_tex_to_color_map({ + "Line": PINK, "scalar":BLUE, "area":TEAL_A + }) + area_text.to_edge(TOP,buff=MED_SMALL_BUFF) + + self.remove(self.values_on_line_text) + self.add_fixed_in_frame_mobjects(area_text) + self.play(Write(area_text)) + self.play(Write(self.area),run_time=2) + self.play(FadeOut(VGroup(surface,fn_text))) + self.wait() + + self.stop_ambient_camera_rotation() + # self.get_lines() + + self.remove(axes,surface) + self.trasform_to_graphs() + self.wait(2) + + + + + def get_line_of_int(self): + line_of_int_text=TextMobject(r"Line of integration is\\","$\\vec r(t)=\cos(t)\hat x+\sin(t)\hat y$") + line_of_int_text[1].set_color(PINK) + line_of_int_text.to_edge(TOP,buff=SMALL_BUFF) + + + line_of_int=(self.get_curve( + self.Func,on_surface=False + )) + line_of_int.set_style( + stroke_width=5, + stroke_color=PINK, + ) + + self.add_fixed_in_frame_mobjects(line_of_int_text) + self.play(Write(line_of_int_text)) + self.wait() + self.play(ShowCreation(line_of_int),run_time=3) + # self.add(line_of_int) + + self.line_of_int=line_of_int + self.line_of_int_text=line_of_int_text + + def get_field_values_on_line(self): + self.remove(self.line_of_int_text) + + values_on_line_text=TextMobject("Values"," of"," function","on the ","line") + values_on_line_text.set_color_by_tex_to_color_map({ + "Values":YELLOW, "function":BLUE,"line":PINK + }) + values_on_line_text.to_edge(TOP,buff=SMALL_BUFF) + + values_on_surface=(self.get_curve( + self.Func,on_surface=True + )) + values_on_surface.set_style( + stroke_width=5, + stroke_color=YELLOW, + ) + + self.add_fixed_in_frame_mobjects(values_on_line_text) + self.play(Write(values_on_line_text)) + # self.wait() + self.play(ShowCreation(values_on_surface),run_time=3) + # self.add(values_on_surface) + + self.values_on_surface=values_on_surface + self.values_on_line_text=values_on_line_text + + + def trasform_to_graphs(self): + on_surface_graph=(self.get_graph( + self.Func,on_surface=True + )) + on_surface_graph.set_style( + stroke_width=5, + stroke_color=YELLOW, + ) + + line_graph=(self.get_graph( + self.Func,on_surface=False + )) + line_graph.set_style( + stroke_width=5, + stroke_color=PINK, + ) + + self.on_surface_graph=on_surface_graph + self.line_graph=line_graph + graph_area=self.get_area(graph=True) + + into_graph=[ + ReplacementTransform( + self.values_on_surface, + on_surface_graph + ), + ReplacementTransform( + self.line_of_int, + line_graph + ), + ReplacementTransform( + self.area, + graph_area + ), + ] + + self.move_camera( + # distance=20, + phi=90 * DEGREES, + theta=-90 * DEGREES, + added_anims=into_graph, + run_time=2 + ) + + def get_area(self,graph=False): + axes=self.axes + if graph: + on_surface=self.on_surface_graph + on_base=self.line_graph + else: + on_surface=self.values_on_surface + on_base=self.line_of_int + area =Polygon( + *[ + on_surface.get_point_from_function(t) + for t in np.arange(0,PI,0.01) + ], + *[ + on_base.get_point_from_function(t) + for t in np.arange(PI,0,-0.01) + ], + stroke_width=0, + fill_color=TEAL_A, + fill_opacity=.6, + ) + + return area + + def get_curve(self,func,on_surface=False ,**kwargs): + config = dict() + config.update(self.default_graph_style) + config.update({ + "t_min": 0, + "t_max": PI, + }) + config.update(kwargs) + r=abs(self.axes.a) + curve=ParametricFunction( + lambda t: self.axes.c2p( + r*np.cos(t), + r*np.sin(t), + func(r*np.cos(t), r*np.sin(t))*bool(on_surface) + ), + **config, + ) + return curve + + + def get_surface(self, func, **kwargs): + axes=self.axes + config = { + "u_min": axes.a-.2, + "u_max": axes.b+.2, + "v_min": axes.c-.1, + "v_max": axes.d, + "resolution": ( + 2*(axes.y_max - axes.y_min) // axes.y_axis.tick_frequency, + (axes.x_max - axes.x_min) // axes.x_axis.tick_frequency, + ), + } + + config.update(self.default_surface_config) + config.update(kwargs) + return ParametricSurface( + lambda x,y : axes.c2p( + x, y, func(x, y) + ), + **config + ) + + def get_graph(self,func,on_surface=False ,**kwargs): + config = dict() + config.update(self.default_graph_style) + config.update({ + "t_min": 0, + "t_max": PI, + }) + config.update(kwargs) + slice_curve=ParametricFunction( + lambda t: self.axes.c2p( + 4*np.cos(t), + 0, + 2+func(3*np.cos(t), 3*np.sin(t))*bool(on_surface) + ), + **config, + ) + return slice_curve + + def get_lines(self): + pass + axes = self.axes + labels=[axes.x_axis.n2p(axes.a), axes.x_axis.n2p(axes.b), axes.y_axis.n2p(axes.c), + axes.y_axis.n2p(axes.d)] + + + surface_corners=[] + for x,y,z in self.region_corners: + surface_corners.append([x,y,self.Func(x,y)]) + + lines=VGroup() + for start , end in zip(surface_corners, + self.region_corners): + lines.add(self.draw_lines(start,end,"PINK")) + + for start , end in zip(labels, + self.region_corners): + # lines.add(self.draw_lines(start,end,"BLUE")) + # print (start,end) + pass + # self.play(ShowCreation(lines)) + self.add(lines) + + + def draw_lines(self,start,end,color): + start=self.axes.c2p(*start) + end=self.axes.c2p(*end) + line=DashedLine(start,end,color=color) + + return line + + #customize 3D axes + def get_three_d_axes(self, include_labels=True, include_numbers=True, **kwargs): + config = dict(self.axes_config) + config.update(kwargs) + axes = ThreeDAxes(**config) + axes.set_stroke(width=2) + self.axes=axes + + if include_numbers: + self.add_axes_numbers(axes) + + if include_labels: + self.add_axes_labels(axes) + + # Adjust axis orientation + axes.x_axis.rotate( + -90 * DEGREES, LEFT, + about_point=axes.c2p(0, 0, 0), + ) + axes.y_axis.rotate( + 90 * DEGREES, UP, + about_point=axes.c2p(0, 0, 0), + ) + + # Add xy-plane + input_plane = self.get_surface( + lambda x, t: 0 + ) + '''input_plane.set_style( + fill_opacity=0.3, + fill_color=PINK, + stroke_width=.2, + stroke_color=WHITE, + )''' + + axes.input_plane = input_plane + + self.region_corners=[ + input_plane.get_corner(pos) for pos in (DL,DR,UR,UL)] + + return axes + + + def setup_axes(self): + axes = self.get_three_d_axes(include_labels=True) + axes.add(axes.input_plane) + axes.scale(1) + # axes.center() + axes.shift(axes.axes_shift) + + self.add(axes) + self.axes = axes + + def add_axes_numbers(self, axes): + x_axis = axes.x_axis + y_axis = axes.y_axis + tex_vals_x = [ + + ("1", axes.b), + ("-1", axes.a), + ] + tex_vals_y=[ + + ("1", axes.d) + ] + x_labels = VGroup() + y_labels = VGroup() + for tex, val in tex_vals_x: + label = TexMobject(tex) + label.scale(1) + label.next_to(x_axis.n2p(val), DOWN) + # label.rotate(180 * DEGREES) + x_labels.add(label) + x_axis.add(x_labels) + x_axis.numbers = x_labels + + for tex, val in tex_vals_y: + label = TexMobject(tex) + label.scale(1) + label.next_to(y_axis.n2p(val), LEFT) + label.rotate(90 * DEGREES) + y_labels.add(label) + + y_axis.add(y_labels) + y_axis.numbers = y_labels + + return axes + + def add_axes_labels(self, axes): + x_label = TexMobject("x") + x_label.next_to(axes.x_axis.get_end(), RIGHT) + axes.x_axis.label = x_label + + y_label = TextMobject("y") + y_label.rotate(90 * DEGREES, OUT) + y_label.next_to(axes.y_axis.get_end(), UP) + axes.y_axis.label = y_label + + z_label = TextMobject("z") + z_label.rotate(90 * DEGREES, RIGHT) + z_label.next_to(axes.z_axis.get_zenith(), LEFT) + axes.z_axis.label = z_label + for axis in axes: + axis.add(axis.label) + return axes + + + + #uploaded by Somnath Pandit.FSF2020_Line_Integrals + + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file3_vector_line_int_as_sum.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file3_vector_line_int_as_sum.gif Binary files differnew file mode 100644 index 0000000..46b35bc --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file3_vector_line_int_as_sum.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file3_vector_line_int_as_sum.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file3_vector_line_int_as_sum.py new file mode 100644 index 0000000..78294cc --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file3_vector_line_int_as_sum.py @@ -0,0 +1,326 @@ +from manimlib.imports import * + + +class LineIntegrationAsSum(GraphScene): + CONFIG = { + "x_min" : 0, + "x_max" : 10, + "y_min" : 0, + "y_max" : 6, + "graph_origin": ORIGIN+5*LEFT+3*DOWN, + "x_axis_width": 10, + "y_axis_height": 6 , + "x_tick_frequency": 2, + "y_tick_frequency": 2, + "Func":lambda x : 1+x**1.3*np.exp(-.12*(x-2)**2)*np.sin(x/4), + "a": 1 ,"b": 9, "n": 15, + } + + 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) + + + curve=self.get_graph( + self.Func, + x_min=self.a, + x_max=self.b, + ) + curve.set_color([BLACK,BLUE,BLUE,BLUE,BLACK]) + curve_label= self.get_graph_label( + curve, + label="\\text{path of intgration}", + x_val=4, + direction=UR, + buff=.6, + color=BLUE + ) + self.curve=curve + self.curve_label=curve_label + + self.get_vector_field() + + + self.play(ShowCreation(VGroup(curve,curve_label))) + self.wait(.6) + self.break_in_arcs() + self.show_the_sum() + + self.wait(2) + + + def get_vector_field(self): + func = lambda v: np.array([ + v[0], # x + -v[1], # y + 0 # z + ]) + vector_field= VectorField( + func, + delta_x=1, + delta_y=1, + colors=[GREEN_A,GREEN_C], + length_func= lambda norm: .8*sigmoid(norm), + vector_config={ + "stroke_width": 2 + } + ) + + self.vector_field= vector_field + + + def break_in_arcs(self): + + self.write_about_breaking() + + dl=0.8 + self.get_breakers(dl) + self.wait(2) + self.play(FadeOut(self.upto_break_text)) + self.dl=dl + + def write_about_breaking(self): + breaking_text=TextMobject("\\texttt{..broken}"," into small", "subarcs") + breaking_text.set_color_by_tex_to_color_map({ + "broken":RED,"subarcs": BLUE + }) + breaking_text.next_to(self.curve_label,DOWN) + breaking_text.align_to(self.curve_label,LEFT) + self.play( + Write(breaking_text) + ) + + self.upto_break_text=VGroup( + self.curve_label, + breaking_text, + ) + + def get_breakers(self,dl): + point=self.a + points=[] + while point<(self.b-dl) : + start=point + end=point+dl + points += [end] + breaker=Line( + self.input_to_graph_point(start,self.curve), + self.input_to_graph_point(end,self.curve), + stroke_width=2, + color=RED, + ) + breaker.rotate(PI/2).scale(.5) + + point=end + self.play(FadeIn(breaker),run_time=.2) + # self.add(breaker) + + del points[-1] + self.points=points + + + def show_the_sum(self): + at_any_points_text=TextMobject("At any ","point", "in each ", "subarc") + at_any_points_text.set_color_by_tex_to_color_map({ + "point":YELLOW , "subarc": BLUE + }) + at_any_points_text.to_edge(TOP,buff=SMALL_BUFF) + + evaluate_text=TextMobject("$\\vec F(x,y)$ ", "is evaluated").next_to(at_any_points_text,DOWN) + evaluate_text.set_color_by_tex("$\\vec F(x,y)$",ORANGE) + + multiply_text=TextMobject("...is multiplied with ","$\\Delta s_i$") + multiply_text.set_color_by_tex("\\Delta s_i", BLUE) + multiply_text.next_to(at_any_points_text,DOWN) + + + + self.at_any_points_text=at_any_points_text + self.evaluate_text=evaluate_text + self.multiply_text=multiply_text + + dots=[] + for point in self.points: + + dot=Dot( + point=self.input_to_graph_point(point,self.curve), + radius= .7*DEFAULT_DOT_RADIUS, + stroke_width= 0, + fill_opacity= 1.0, + color= YELLOW, + ) + dots+=[dot] + + self.play( + Write(at_any_points_text), + FadeIn(VGroup(*dots)),run_time=1.5 + ) + self.dots=dots + + self.wait() + self.show_the_dot_product() + self.multiply_with_ds() + self.construct_equation() + + + def show_the_dot_product(self): + index=-(len(self.points)//3) + self.index=index + + dot=self.dots[index] + + + dot_prod_text=TextMobject("Dot Product of", "$\\vec F(x_i,y_i)$", "and","$\\vec T(x_i,y_i)$") + dot_prod_text.set_color_by_tex_to_color_map({ + "\\vec F(x_i,y_i)":ORANGE , + "\\vec T(x_i,y_i)": "#DC75CD" , + }) + dot_prod_text.to_edge(TOP,buff=SMALL_BUFF) + + + point_coord=TextMobject("$(x_i,y_i)$",color=YELLOW) + point_coord.next_to(dot,DL,buff=.01).scale(.8) + + func_val=TextMobject("$\\vec F(x_i,y_i)$",color=ORANGE) + func_val.next_to(dot,UR).scale(.8) + + self.dot_prod_text=dot_prod_text + self.func_val=func_val + + dot.set_color(ORANGE).scale(1.2) + + + self.play(FadeIn(VGroup(point_coord,dot))) + self.play(Write(self.evaluate_text)) + self.wait(1) + self.play(FadeOut(self.vector_field)) + self.get_vector_and_tangent() + self.dot_product() + + + self.wait(2) + self.remove(point_coord) + + + def get_vector_and_tangent(self): + dot=self.dots[self.index] + self.show_specific_vectors(dot) + self.play(Write(self.func_val)) + self.wait(1) + self.show_tangent(dot) + self.play(FadeIn(VGroup(*[ + dot.set_color(ORANGE).scale(1.4) + for dot in self.dots ] + ))) + + + def show_specific_vectors(self,dots): + for dot in dots: + vector=self.vector_field.get_vector(dot.get_center()) + vector.set_color(ORANGE) + + self.play(Write(vector),run_time=.2) + + + def show_tangent(self,dot): + tangent_sym=TextMobject("$\\vec T(x_i,y_i)$",color="#DC75CD").scale(.8) + x=dot.get_center() + angle=self.angle_of_tangent( + self.point_to_coords(x)[0], + self.curve, + dx=0.01 + ) + vect = Vector().rotate(angle,about_point=x) + vect.set_color("#DC75CD") + tangent=vect.next_to(x,DR,buff=0) + tangent_sym.next_to(tangent,DOWN,buff=.1) + self.play(Write(VGroup(tangent,tangent_sym))) + + self.tangent_sym=tangent_sym + + def dot_product(self): + + dot_sym=Dot().next_to(self.func_val,RIGHT) + + self.play(FadeOut(VGroup( + self.at_any_points_text, + self.evaluate_text + ))) + self.play(Write(self.dot_prod_text)) + self.play( + FadeIn(dot_sym), + ApplyMethod( + self.tangent_sym.next_to, + dot_sym, RIGHT + )) + + self.dot_sym=dot_sym + + def multiply_with_ds(self): + self.get_ds() + + self.play(GrowFromCenter(self.ds_brace_group)) + self.wait(2) + self.play(Write(self.multiply_text)) + self.play(ApplyMethod( + self.ds_brace_label.next_to, + self.tangent_sym, RIGHT,buff=.15 + )) + + + + def get_ds(self): + p1= self.dots[self.index] + p2= self.dots[self.index+1] + ds_brace=Brace(VGroup(p1,p2),DL) + ds_brace.move_to(p1,UR) + ds_brace_label=ds_brace.get_text("$\Delta s_i$", buff = .05) + ds_brace_label.set_color(BLUE) + self.ds_brace=ds_brace + self.ds_brace_label=ds_brace_label + self.ds_brace_group=VGroup(ds_brace,ds_brace_label) + + + def construct_equation(self): + sum_up_text=TextMobject("and"," summed ", "for all i' s") + sum_up_text.set_color_by_tex("summed",PURPLE_A) + sum_up_text.next_to(self.multiply_text,DOWN,buff=MED_SMALL_BUFF) + sum_up_text.shift(LEFT) + + sum_eqn=TextMobject("$$\\sum_i^{ } $$").set_color(PURPLE_A) + sum_eqn.move_to(self.graph_origin+6.5*self.X+4*self.Y) + + line_integral_text=TextMobject("The Value of the"," line ","integral is").to_edge(TOP,buff=MED_SMALL_BUFF) + line_integral_text.set_color_by_tex("line",BLUE_C) + approx=TextMobject("$\\approx$",color=RED).next_to(sum_eqn,LEFT) + multipled=VGroup( + self.func_val, + self.dot_sym, + self.tangent_sym, + self.ds_brace_label + ) + + + self.play(Write(sum_up_text)) + self.show_specific_vectors(self.dots) + self.play(FadeIn(sum_eqn)) + self.play(ApplyMethod( + multipled.next_to,sum_eqn,RIGHT + )) + self.wait() + self.play(FadeOut(VGroup( + self.dot_prod_text, + self.multiply_text, + sum_up_text + ))) + self.play(Write(line_integral_text)) + self.play(FadeIn(approx)) + + + +#uploaded by Somnath Pandit.FSF2020_Line Integrals + + + diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file4_helix.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file4_helix.gif Binary files differnew file mode 100644 index 0000000..ceedb1f --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file4_helix.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file4_helix.py b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file4_helix.py new file mode 100644 index 0000000..50aeb33 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/line-integrals/file4_helix.py @@ -0,0 +1,245 @@ +from manimlib.imports import * + +class ParametricCurve(ThreeDScene): + + CONFIG = { + "axes_config": { + "x_min": 0, + "x_max": 3, + "y_min": 0, + "y_max": 3, + "z_min": 0, + "z_max": 4, + "a":0 ,"b": 2, "c":0 , "d":2, + "axes_shift":2*IN+1.4*RIGHT+1.4*DOWN, + "x_axis_config": { + "tick_frequency": 1, + "include_tip": False, + }, + "y_axis_config": { + "tick_frequency": 1, + "include_tip": False, + }, + "z_axis_config": { + "tick_frequency": 1, + # "include_tip": False, + }, + }, + + } + + + def construct(self): + + self.setup_axes() + + self.set_camera_orientation( + distance=25, + phi=60 * DEGREES, + theta=40 * DEGREES, + ) + + label=TextMobject("Helix",color=PURPLE).scale(1.6) + label.to_corner(UR,buff=2) + self.add_fixed_in_frame_mobjects(label) + + helix=self.get_helix( + radius=1.5, + t_min= 0, + t_max= 4*PI, + color=PURPLE + ) + parameter_label=TextMobject( + "Parametric equation: ", + color=TEAL + ).next_to(label,DOWN,buff=.3 + ) + parametric_eqn=TextMobject( + "$x=\cos$ (","t", + r")\\$y=\sin $(","t", + r")\\$z$=","t" + ).next_to(parameter_label,DOWN,buff=.1) + parametric_eqn.set_color_by_tex("t",RED) + self.parametric_eqn=parametric_eqn + + parametriztion=VGroup( + parameter_label, + parametric_eqn + ) + + + self.play(ShowCreation(helix),run_time=2) + self.begin_ambient_camera_rotation(.1) + self.wait(1) + self.add_fixed_in_frame_mobjects(parametriztion) + self.play(Write(parametriztion)) + self.wait(1) + self.stop_ambient_camera_rotation() + self.move_camera( + distance=20, + phi=85 * DEGREES, + # theta=-90 * DEGREES, + run_time=3 + ) + scale_axes=VGroup(self.axes,helix).scale(1.2) + self.show_the_parameter() + self.wait(2) + + + + def get_helix(self,radius=1, **kwargs): + config = { + "t_min": 0, + "t_max": 2*PI, + } + config.update(kwargs) + helix= ParametricFunction( + lambda t : self.axes.c2p( + radius*np.cos(t), + radius*np.sin(t), + t/4 + ), + **config + ) + + self.helix=helix + return helix + + def show_the_parameter(self): + t_tracker = ValueTracker(0) + t=t_tracker.get_value + + t_label = TexMobject( + "t = ",color=RED + ).next_to(self.parametric_eqn,DL,buff=.85) + + t_text = always_redraw( + lambda: DecimalNumber( + t(), + color=GOLD, + ).next_to(t_label, RIGHT, MED_SMALL_BUFF) + ) + t_text.suspend_updating() + + dot = Sphere( + radius= 1.5*DEFAULT_DOT_RADIUS, + stroke_width= 1, + fill_opacity= 1.0, + ) + dot.set_color(GOLD) + dot.add_updater(lambda v: v.move_to( + self.helix.get_point_from_function(PI*t()) + )) + + pi = TexMobject( + "\\pi ", + color=GOLD, + ).next_to(t_text,RIGHT,buff=-.3) + + group = VGroup(t_text,t_label,pi).scale(1.5) + + self.wait(1) + self.add_fixed_in_frame_mobjects(group) + t_text.resume_updating() + self.play(FadeIn(group)) + self.add(dot) + self.play( + t_tracker.set_value,2, + rate_func=linear, + run_time=5 + ) + + +#-------------------------------------------------------- + + #customize 3D axes + def get_three_d_axes(self, include_labels=True, include_numbers=False, **kwargs): + config = dict(self.axes_config) + config.update(kwargs) + axes = ThreeDAxes(**config) + axes.set_stroke(width=1.5) + + if include_numbers: + self.add_axes_numbers(axes) + + if include_labels: + self.add_axes_labels(axes) + + # Adjust axis orientation + axes.x_axis.rotate( + 90 * DEGREES, LEFT, + about_point=axes.c2p(0, 0, 0), + ) + axes.y_axis.rotate( + 90 * DEGREES, UP, + about_point=axes.c2p(0, 0, 0), + ) + + + return axes + + + def setup_axes(self): + axes = self.get_three_d_axes(include_labels=True) + axes.scale(1) + # axes.center() + axes.shift(axes.axes_shift) + self.add(axes) + self.axes = axes + + def add_axes_numbers(self, axes): + x_axis = axes.x_axis + y_axis = axes.y_axis + tex_vals_x = [ + ("1", axes.b), + ] + tex_vals_y=[ + ("1", axes.d) + ] + x_labels = VGroup() + y_labels = VGroup() + for tex, val in tex_vals_x: + label = TexMobject(tex) + label.scale(1) + label.next_to(x_axis.n2p(val), DOWN) + label.rotate(180 * DEGREES) + x_labels.add(label) + x_axis.add(x_labels) + x_axis.numbers = x_labels + + for tex, val in tex_vals_y: + label = TexMobject(tex) + label.scale(1) + label.next_to(y_axis.n2p(val), LEFT) + label.rotate(90 * DEGREES) + y_labels.add(label) + + y_axis.add(y_labels) + y_axis.numbers = y_labels + + return axes + + def add_axes_labels(self, axes): + x_label = TexMobject("x") + x_label.next_to(axes.x_axis.get_end(), RIGHT) + axes.x_axis.label = x_label + + y_label = TextMobject("y") + y_label.rotate(90 * DEGREES, OUT) + y_label.next_to(axes.y_axis.get_end(), UP) + axes.y_axis.label = y_label + + z_label = TextMobject("z") + z_label.rotate(90 * DEGREES, LEFT) + z_label.next_to(axes.z_axis.get_zenith(), LEFT) + axes.z_axis.label = z_label + for axis in axes: + axis.add(axis.label) + return axes + + #uploaded by Somnath Pandit.FSF2020_Line_integrals + + + + + |
