importing changes

FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/README.md

@@ -1,20 +1,26 @@
 <h1><div align=”center”><b>SubTopic: Lagrange Multipliers</b></h1></div>
 <br/></br>
 
-<tab>file1_Extrema_over_g(x,y)=k
+<tab>file1_Extrema_over_g(x,y)
+
+![file1_Extrema_over_g(x,y)](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x%2Cy)%3Dk.gif?raw=true)
+<br/></br>
+<br/></br>
 
-![file1_Extrema_over_g(x,y)=k](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x%2Cy)%3Dk.gif?raw=true)
+<tab>file2_Constraint_circle_with_contour_plot_of_the_surface_x^2+y^2+x^3-y^3
+
+ ![file2_Constraint_circle_with_contour_plot_of_the_surface_x^2+y^2+x^3-y^3](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file2_Constraint_circle_with_contour_plot_of_the_surface_x%5E2%2By%5E2%2Bx%5E3-y%5E3.gif?raw=true)
 <br/></br>
 <br/></br>
 
-<tab>file2_Geometric_Proof
+<tab>file3_Geometric_Proof
 
-![file2_Geometric_Proof](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file2_Geometric_Proof.gif?raw=true)
+![file3_Geometric_Proof](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Geometric_Proof.gif?raw=true)
 <br/></br>
 <br/></br>
 
-<tab>file3_Constraints_g_and_h
+<tab>file4_Constraints_g_and_h
 
-![file3_Constraints_g_and_h](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Constraints_g_and_h.gif?raw=true)
+![file4_Constraints_g_and_h](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file4_Constraints_g_and_h.gif?raw=true)
 <br/></br>
 <br/></br>

FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.py

@@ -4,10 +4,10 @@
 #---- optimizing funtion f(x,y) w.r.t to g(x,y)
 class ConstrainedExtrema(ThreeDScene):
     def construct(self):
-        axes = ThreeDAxes().fade(0.4)        
-        label_x = TextMobject("$x$").shift([5.5,-0.5,0])  #---- x axis
-        label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis
-
+        axes = ThreeDAxes().scale(0.7).rotate(math.radians(180))       
+        label_x = TextMobject("$x$").shift(4*LEFT).fade(0.4)  #---- x axis
+        label_y = TextMobject("$y$").shift(3.2*DOWN+0.2*RIGHT).rotate(math.radians(180)).fade(0.4)  #---- y axis
+        
         surface = ParametricSurface(
             lambda u, v: np.array([
                 u,

FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file2_Constraint_circle_with_contour_plot_of_the_surface_x^2+y^2+x^3-y^3.py

@@ -0,0 +1,125 @@
+from manimlib.imports import*
+import math as m
+
+class Contourplots(ThreeDScene):
+    def construct(self):
+        axes = ThreeDAxes().scale(0.7).rotate(math.radians(180)).fade(0.6)     
+        label_x = TextMobject("$x$").shift(4*LEFT).fade(0.4)  #---- x axis
+        label_y = TextMobject("$y$").shift(3.5*DOWN+0.2*RIGHT).rotate(math.radians(180)).fade(0.4)  #---- y axis
+
+        #---- surface of the function
+        surface = ParametricSurface(
+            lambda u, v: np.array([
+                u,
+                v,
+                u**2+v**2+u**3-v**3
+            ]),u_min=-0.5,u_max=0.5, v_min=-0.5,v_max=0.5).scale(6).shift([0,0,2.5]).set_color(TEAL).fade(0.5)
+
+        #---- contour plots 
+        curve1 = ParametricSurface(
+            lambda u, v: np.array([
+                u*0.4,
+                m.exp(m.sin(v**2)),
+                -v*2.2
+            ]),v_min =-1.3 , v_max =1.3, u_min = -0.1, u_max = 0.1).scale(0.8).shift([0,-2.65,0]).set_color("#800000").rotate(1.2,DOWN)
+
+        curve2 = ParametricSurface(
+            lambda u, v: np.array([
+                u*0.4,
+                m.exp(m.sin(v**2)),
+                -v*2.2
+            ]),v_min =-1.3 , v_max =1.3, u_min = -0.1, u_max = 0.1).scale(0.9).shift([0,-3,0]).set_color("#800000").rotate(1.2,DOWN)
+
+        curve3 = ParametricSurface(
+            lambda u, v: np.array([
+                u*0.4,
+                m.exp(m.sin(v**2)),
+                -v*2.2
+            ]),v_min =-1.3 , v_max =1.3 , u_min = -0.1, u_max = 0.1).scale(1).shift([0,-3.5,0]).set_color("#800000").rotate(1.2,DOWN)
+
+        curve4 = ParametricSurface(
+            lambda u, v: np.array([
+                u*0.4,
+                m.exp(m.sin(v**2)),
+                -v*2.2
+            ]),v_min =-1.3 , v_max =1.3 , u_min = -0.1, u_max = 0.1).scale(1.1).shift([0,-4,0]).set_color("#800000").rotate(1.2,DOWN)
+
+        curve5 = ParametricSurface(
+            lambda u, v: np.array([
+                u*0.4,
+                -m.exp(m.sin(v**2)),
+                v*2.2
+            ]),v_min =-1.3 , v_max =1.3, u_min = -0.1, u_max = 0.1).scale(0.9).shift([0,3,0]).set_color("#800000").rotate(1.2,DOWN)
+
+        curve6 = ParametricSurface(
+            lambda u, v: np.array([
+                u*0.4,
+                -m.exp(m.sin(v**2)),
+                v*2.2
+            ]),v_min =-1.3 , v_max =1.3, u_min = -0.1, u_max = 0.1).scale(1).shift([0,3.5,0]).set_color("#800000").rotate(1.2,DOWN)
+
+        curve7 = ParametricSurface(
+            lambda u, v: np.array([
+                u*0.4,
+                -m.exp(m.sin(v**2)),
+                v*2.2
+            ]),v_min =-1.3 , v_max =1.3, u_min = -0.1, u_max = 0.1).scale(1.1).shift([0,4,0]).set_color("#800000").rotate(1.2,DOWN)
+
+        curve8 = ParametricSurface(
+            lambda u, v: np.array([
+                u*0.4,
+                -m.exp(m.sin(v**2)),
+                v*2.2
+            ]),v_min =-1.3 , v_max =1.3, u_min = -0.1, u_max = 0.1).scale(0.8).shift([0,2.65,0]).set_color("#800000").rotate(1.2,DOWN)
+
+        #---------------
+
+        #---- label contours
+        label_c1 = TextMobject("4",color = '#FFA074').shift([0.2,3,0.5]).scale(0.5).rotate(m.radians(180))
+        label_c2 = TextMobject("3",color = '#FFA074').shift([0.2,2.4,0.5]).scale(0.5).rotate(m.radians(180))
+        label_c3 = TextMobject("2",color = '#FFA074').shift([0.2,1.9,0.5]).scale(0.5).rotate(m.radians(180))
+        label_c4 = TextMobject("1",color = '#FFA074').shift([0.2,1.4,0.5]).scale(0.5).rotate(m.radians(180))
+        label_c5 = TextMobject("1",color = '#FFA074').shift([0.2,-1.5,0.5]).scale(0.5).rotate(m.radians(180))
+        label_c6 = TextMobject("2",color = '#FFA074').shift([0.2,-2,0.5]).scale(0.5).rotate(m.radians(180))
+        label_c7 = TextMobject("3",color = '#FFA074').shift([0.2,-2.5,0.5]).scale(0.5).rotate(m.radians(180))
+        label_c8 = TextMobject("4",color = '#FFA074').shift([0.2,-3,0.5]).scale(0.5).rotate(m.radians(180))
+
+        contourplot = VGroup(curve1,curve2,curve3,curve4,curve5,curve6,curve7,curve8,label_c1,label_c2,label_c3,label_c4,label_c5,label_c6,label_c7,label_c8)
+
+
+        circle = Circle(color='#FF00FF',fill_opacity=0.3).shift([-0.5,-1.2,1.5]).rotate(1.9,UP).scale(0.8)
+        circle2 = ParametricSurface(
+            lambda u, v: np.array([
+                1*np.sin(u)*np.cos(v),
+                1*np.sin(u)*np.sin(v),
+                -1*np.sin(u)*np.sin(u)+2
+            ]),u_min=0,u_max=PI/2,v_min=0,v_max=2*PI).set_color('#FF00FF').scale(0.6).shift([1.2,2,-3]).scale(1.24)
+
+        maxima = Dot(color = '#4169E1').shift([0.7,1.2,1.5])
+        minima  = Dot(color = '#4169E1').shift([1.4,2.2,1.5])
+
+        min_text = TextMobject("minimum over $g(x,y)=k$",color = '#FFA074').scale(0.6).shift([-2.2,-1,1.5])
+        max_text = TextMobject("maximum over $g(x,y)=k$",color = '#FFA074').shift([-2.3,-2.6,1.5]).scale(0.6)
+
+
+        self.set_camera_orientation(phi=75 * DEGREES, theta = 45*DEGREES)
+        self.add(axes)        
+        self.add(label_x) 
+        self.add(label_y)
+        self.wait(1)
+        self.play(Write(surface))
+        self.play(Write(circle))
+        self.wait(1)
+        self.play(FadeOut(circle))
+        self.wait(1)
+        self.move_camera(phi=0 * DEGREES, theta = 90*DEGREES)
+        self.wait(1)        
+        self.add(contourplot)
+        self.play(FadeOut(surface))
+        self.wait(1)
+        self.add(circle2)
+        self.wait(1)
+        self.play(Write(minima),Write(maxima)) 
+        self.add_fixed_in_frame_mobjects(max_text) 
+        self.add_fixed_in_frame_mobjects(min_text) 
+        self.wait(1)

FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file4_Constraints_g_and_h.py

@@ -0,0 +1,52 @@
+from manimlib.imports import*
+import math as m
+
+    class Constraints(ThreeDScene):
+    def construct(self):
+       axes = ThreeDAxes().rotate(m.radians(75))
+        label_x = TextMobject("$x$").shift([-5.5,1,0]).fade(0.4)  #---- x axis
+        label_y = TextMobject("$y$").shift([1,5.5,0]).rotate(-4.5).fade(0.4)  #---- y axis
+
+        cylinder = ParametricSurface(
+            lambda u, v: np.array([
+                np.cos(TAU * u),
+                np.sin(TAU * u),
+                2 * (1-1.5*v)
+            ]),checkerboard_colors=[YELLOW_C,YELLOW_D,YELLOW_E]).shift([0.5,0.5,-0.13]).scale(1) 
+
+        plane =  ParametricSurface(
+            lambda u, v: np.array([
+                u,
+                v,
+                u+v
+            ]),checkerboard_colors=[TEAL_C,TEAL_D,TEAL_E]).shift([0,0,0]).rotate(m.radians(-40),RIGHT).scale(4).fade(0.3)
+
+        c = Circle(color='#FF00FF',fill_opacity=0.3).shift([0.7,-1.3,0.4]).rotate(2.5,UP).scale(1.32)
+
+        f_text = TextMobject("$f(x,y)=x^2+y^2+z^2$",color = '#FFA074').scale(0.6).to_corner(UL)
+        g_text = TextMobject("$g(x,y)=x^2+y^2+1$",color = '#FFA074').scale(0.6).to_corner(UL)
+        h_text = TextMobject("$h(x,y)=x+y-z=1$",color = '#FFA074').scale(0.6).to_corner(UL)
+
+
+
+        self.set_camera_orientation(phi=65*DEGREES,theta=95*DEGREES)
+
+        self.add(axes)        
+        self.add(label_x) 
+        self.add(label_y)
+        self.wait(1) 
+        self.add_fixed_in_frame_mobjects(f_text)    
+        self.play(Write(c))
+        self.wait(1)
+        self.play(FadeOut(f_text))
+        self.wait(1)
+        self.add_fixed_in_frame_mobjects(g_text)
+        self.play(Write(cylinder)) 
+        self.wait(1)
+        self.play(FadeOut(g_text))
+        self.wait(1)
+        self.add_fixed_in_frame_mobjects(h_text)
+        self.play(Write(plane))            
+        self.wait(1)
+        self.play(FadeOut(h_text))
+        self.wait(1)

FSF-2020/calculus-of-several-variables/div-curl-grad-and-all-that/Conservative fields/Conservative_file1_paths-3d.py

@@ -0,0 +1,41 @@
+from manimlib.imports import *
+import numpy as np
+
+class routes(ThreeDScene):
+	def construct(self):
+
+		axes = ThreeDAxes()
+		self.begin_ambient_camera_rotation(rate=0.08)
+		self.set_camera_orientation(phi=45 * DEGREES,theta=-65*DEGREES, distance = 45)
+		function = ParametricFunction(
+            lambda t: np.array([
+            1.5*np.cos(t),
+            1.5*np.sin(t),
+            t/4
+            ]), t_min = 0, t_max =3*PI).set_color("#0EB1D2")
+
+
+		line = Line(np.array([1.5, 0, 0]), np.array([-1.5, 0, 2.35619])).set_color("#F8F32B")
+
+		dot1 = Sphere(radius = 0.1).move_to(np.array([1.5,0,0])).set_color("#74226C")
+		dot2 = Sphere(radius = 0.1).move_to(np.array([1.5,0,0])).set_color("#74226C")
+
+		label1 = TexMobject(r"A").move_to(np.array([1.5,0,0])).set_color("#FCF7F8")
+		label2 = TexMobject(r"B").move_to(np.array([-1.5,0,2.42])).set_color("#FCF7F8")
+
+		title = TexMobject(r"\textit{Work done}",r"\textit{ against gravity is the}", r"\textit{ same for both paths}").set_color("#F1E3F3").move_to(np.array([0,-3,0]))
+		title[1].set_color("#F8F32B")
+
+		self.add(axes)
+		self.wait()
+		self.play(ShowCreation(function), ShowCreation(line), run_time = 2)
+		self.wait()
+		self.play(MoveAlongPath(dot1, function), run_time = 2)
+		self.wait()
+		self.play(MoveAlongPath(dot2, line), run_time = 1.2)
+		self.wait()
+		self.add_fixed_in_frame_mobjects(title)
+		self.play(Write(title))
+		self.wait()
+
+