diff options
Diffstat (limited to 'FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral')
7 files changed, 649 insertions, 0 deletions
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..3cdddae --- /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.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..fd3d9b5 --- /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 +1,317 @@ +from manimlib.imports import * + +class GradOfScalarFunc(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": 5, + "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": 3, + "colors": [TEAL,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": .2, + "stroke_color": WHITE, + "stroke_opacity": 0.5, + }, + } + + + 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)=","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").set_color(GOLD) + grad_text.next_to(scalar_fn_text,DOWN).scale(.9) + + VGroup( + grad_text[0][1], + grad_text[0][9] + ).set_color(BLUE) + VGroup( + grad_text[0][5:8], + grad_text[0][13:16] + ).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) + + + show_vects=[ + FadeIn(v_field1), + ] + + self.begin_ambient_camera_rotation(rate=.2) + self.move_camera( + # distance=20, + phi=60 * DEGREES, + added_anims=show_vects, + run_time=4.5 + ) + + self.play(FadeOut(grad_text)) + self.wait(2) + self.stop_ambient_camera_rotation() + + self.add_fixed_in_frame_mobjects(vector_field_text) + vector_field= [ + FadeOut(s_field1), + Write(vector_field_text), + ] + self.move_camera( + # distance=20, + phi=0 * DEGREES, + theta=-90 * DEGREES, + added_anims=vector_field, + 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) + vector_field.move_to(self.axes.c2p(0,0,0)) + self.vector_field=vector_field + + if on_surface: + vector_field=self.get_vectors_on_surface() + + return vector_field + + + + def get_vectors_on_surface(self): + vectors_on_surface = VGroup(*[ + self.vector_field.get_vector(point) + 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.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..b8f7cfa --- /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_f","P_i"] + 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.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/fundamental-theorem-of-line-integral/gifs/file1_grad_of_scalar_function.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/gifs/file1_grad_of_scalar_function.gif Binary files differnew file mode 100644 index 0000000..1fd2e15 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/gifs/file1_grad_of_scalar_function.gif diff --git a/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/gifs/file2_line_int_independent_of_path.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/gifs/file2_line_int_independent_of_path.gif Binary files differnew file mode 100644 index 0000000..8d375bb --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/gifs/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/gifs/file3_line_int_example.gif b/FSF-2020/calculus-of-several-variables/integrals-of-multivariable-functions/fundamental-theorem-of-line-integral/gifs/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/gifs/file3_line_int_example.gif |
