Lagrange Multipliers

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

@@ -9,7 +9,7 @@
 
 <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)
+ ![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>
 

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

@@ -1,47 +1,105 @@
 from manimlib.imports import*
 import math as m
 
-#---- contour plot of the surface with constraint circle
-class ContourScene(ThreeDScene):
+class Contourplots(ThreeDScene):
     def construct(self):
-        axes = ThreeDAxes().scale(0.7).rotate(m.radians(180)).fade(0.6)     
+        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(3.2*DOWN+0.2*RIGHT).rotate(m.radians(180)).fade(0.4)  #---- y 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 f(x,y) = x^2+y^2+x^3-y^3
+        #---- 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(5).shift([0,-0.5,2.5]).set_color(TEAL).fade(0.5)
-
+            ]),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 of the surface of the function
+        #---- 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)
 
-        c0 = Circle(color = '#800000').scale(0.5).shift([0,-0.5,0])
-        c1 = Circle(color = '#800000').scale(1).shift([0,-0.5,0])
-        c2 = Circle(color = '#800000').scale(1.5).shift([0,-0.5,0])
-        c3 = Circle(color = '#800000').scale(2).shift([0,-0.5,0])
-        c4 = Circle(color = '#800000').scale(2.5).shift([0,-0.5,0])
+        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)
 
-        #---- constraint circle
-        circle = Circle(color='#FF00FF',fill_opacity=0.3).shift([-0.5,-1.2,1.5]).rotate(1.9,UP).scale(0.8)
-        circle2 = Circle(color='#FF00FF',fill_opacity=0.3).shift([0.74,0.95,1.5]).rotate(1.9,UP).scale(0.8)
+        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)
 
-        maxima = Dot(color = '#4169E1').shift([0.7,0.15,1.5]) #---- point of maxima
-        minima  = Dot(color = '#4169E1').shift([0.8,1.7,1.5]) #---- point of minima
+        #---------------
+
+        #---- 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)
+
 
-        min_text = TextMobject("minimum over $g(x,y)=k$",color = '#FFA074').scale(0.6).shift([-2,0.16,1.5])
-        max_text = TextMobject("maximum over $g(x,y)=k$",color = '#FFA074').shift([-2.3,-2.6,1.5]).scale(0.6).shift(0.5*UP)
+        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])
 
-        #---- labelling contour curves
-        label_c0 = TextMobject("1",color = '#FFA074').shift([0.2,0.1,0.5]).scale(0.5)
-        label_c1 = TextMobject("2",color = '#FFA074').shift([0.2,-0.6,0.5]).scale(0.5)
-        label_c2 = TextMobject("3",color = '#FFA074').shift([0.2,-1.1,0.5]).scale(0.5)
-        label_c3 = TextMobject("4",color = '#FFA074').shift([0.2,-1.6,0.5]).scale(0.5)
-        label_c4 = TextMobject("5",color = '#FFA074').shift([0.2,-2.1,0.5]).scale(0.5)
+        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)
@@ -56,15 +114,10 @@ def construct(self):
         self.wait(1)
         self.move_camera(phi=0 * DEGREES, theta = 90*DEGREES)
         self.wait(1)        
-        self.play(Write(c0),Write(c1),Write(c2),Write(c3),Write(c4)) 
-        self.play(FadeOut(surface))  
-        self.add_fixed_in_frame_mobjects(label_c0) 
-        self.add_fixed_in_frame_mobjects(label_c1)
-        self.add_fixed_in_frame_mobjects(label_c2)
-        self.add_fixed_in_frame_mobjects(label_c3)
-        self.add_fixed_in_frame_mobjects(label_c4)
+        self.add(contourplot)
+        self.play(FadeOut(surface))
         self.wait(1)
-        self.play(Write(circle2))  
+        self.add(circle2)
         self.wait(1)
         self.play(Write(minima),Write(maxima)) 
         self.add_fixed_in_frame_mobjects(max_text)