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-rw-r--r--FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.gifbin0 -> 595677 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.py57
-rw-r--r--FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file2_Geometric_Proof.py90
-rw-r--r--FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.gifbin0 -> 2177236 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.py29
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diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.gif
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+++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.gif
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diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.py
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+++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file1_Extrema_over_g(x,y)=k.py
@@ -0,0 +1,57 @@
+from manimlib.imports import*
+import math as m
+
+#---- 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]).fade(0.4) #---- x axis
+ label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5).fade(0.4) #---- y axis
+
+ 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([0,1,2.5]).set_color(TEAL).fade(0.4)
+
+ c = Circle().set_color('#FF00FF').shift([-0.4,0,1.5]).rotate(1.9,UP).scale(0.7)
+
+ minima = Dot(color = '#4169E1').shift([-0.5,0.5,1]).rotate(1.571,UP)
+ maxima = Dot(color = '#4169E1').shift([0.1,0,2.2]).rotate(1.571,UP)
+
+ l1 = DashedLine([-0.5,0.5,0.9],[-0.5,0.5,0],color = '#F08080')
+ l2 = DashedLine([0.1,0,2.1],[0.1,0,0],color = '#F08080')
+
+ c2 = Circle(fill_opacity= 0.5).shift([-0.3,0.2,0]).scale(0.4)
+
+ minima_refl = Dot(color = '#4682B4').shift([-0.5,0.5,0]).rotate(1.571,UP)
+ maxima_refl = Dot(color = '#4682B4').shift([0.1,0,0]).rotate(1.571,UP)
+
+ max_text = TextMobject("maximum over $g(x,y)=k$",color = '#FFA074').shift([-1.7,0,0]).scale(0.5).shift(2.2*UP)
+ min_text = TextMobject("minimum over $g(x,y)=k$",color = '#FFA074').shift([2.5,0.5,1]).scale(0.5).shift(0.5*UP)
+ label_f = TextMobject("$z=f(x,y)$",color = '#8A2BE2').scale(0.5).shift(3*UP+3*RIGHT)
+ label_g = TextMobject("$g(x,y)=k$",color = '#8A2BE2').scale(0.5).shift(2*RIGHT)
+
+ self.add(axes)
+ self.add(label_x)
+ self.add(label_y)
+ self.set_camera_orientation(phi=75*DEGREES,theta=45*DEGREES)
+ self.play(Write(surface))
+ self.add_fixed_in_frame_mobjects(label_f)
+ self.wait(2)
+ self.play(Write(c))
+ self.wait(1)
+ self.play(Write(maxima))
+ self.add_fixed_in_frame_mobjects(max_text)
+ self.wait(1)
+ self.play(Write(minima))
+ self.add_fixed_in_frame_mobjects(min_text)
+ self.wait(1)
+ self.play(ShowCreation(l1),ShowCreation(l2))
+ self.play(Write(c2))
+ self.add_fixed_in_frame_mobjects(label_g)
+ self.wait(1)
+ self.play(Write(maxima_refl))
+ self.play(Write(minima_refl))
+ self.wait(1)
diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file2_Geometric_Proof.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file2_Geometric_Proof.py
new file mode 100644
index 0000000..2c2a9de
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file2_Geometric_Proof.py
@@ -0,0 +1,90 @@
+from manimlib.imports import*
+
+#---- visualization of geometric proof of Lagrange multiplier
+class GeometricProof(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes().scale(0.7).rotate(math.radians(180))
+ label_x = TextMobject("$x$").shift(4*RIGHT).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([
+ 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,checkerboard_colors=[GREEN_C, GREEN_E]).scale(1).shift([-1.5,-1.5,0])
+
+ d = Dot([-2,-2.55,0],color = '#800000')
+ a_df = Arrow(color = '#00FFFF').rotate(-2).shift(3.2*DOWN+2.3*LEFT) #---- f parallel to g at maxima
+ a_dg = Arrow(color = '#FF00FF').scale(0.8).shift(3.2*DOWN+2.3*LEFT).rotate(-2) #---- g parallel to f at maxima
+
+ b_dg = Arrow(color = '#00FFFF').rotate(1.1).shift(0.82*LEFT+0.15*UP) #---- g parallel to f at minima
+ b_df = Arrow(color = '#FF00FF').scale(0.6).rotate(-2).shift(1.43*LEFT+1.1*DOWN) #---- f parallel to g at minima
+
+
+ qd = Dot(color = '#800000').shift(1.2*LEFT+0.6*DOWN)
+
+ #---- level curves
+ l1 = Line([-1,-3.1,0],[-4,-3.1,0],color = PINK).rotate(-0.3).fade(0.6)
+ l2 = Line([-0.9,-2.9,0],[-4,-2.9,0],color = PINK).rotate(-0.3).fade(0.6)
+ l3= Line([-0.8,-2.7,0],[-4,-2.7,0],color = PINK).rotate(-0.3).fade(0.6)
+ l4= Line([-0.7,-2.45,0],[-4,-2.45,0],color = PINK).rotate(-0.3).fade(0.6)
+ l5= Line([-0.6,-2.2,0],[-4,-2.25,0],color = PINK).rotate(-0.3).fade(0.6)
+ l6 = Line([-0.5,-2,0],[-4,-2,0],color = PINK).rotate(-0.3).fade(0.6)
+ l7 = Line([-0.4,-1.8,0],[-4,-1.8,0],color = PINK).rotate(-0.3).fade(0.6)
+ l8 = Line([-0.3,-1.6,0],[-4,-1.6,0],color = PINK).rotate(-0.3).fade(0.6)
+ l9= Line([-0.2,-1.4,0],[-4,-1.4,0],color = PINK).rotate(-0.3).fade(0.6)
+ l10= Line([-0.1,-1.2,0],[-4,-1.2,0],color = PINK).rotate(-0.3).fade(0.6)
+ l11 = Line([-0,-1,0],[-4,-1,0],color = PINK).rotate(-0.3).fade(0.6)
+ l12 = Line([-0,-0.8,0],[-4,-0.8,0],color = PINK).rotate(-0.3).fade(0.6)
+ l13= Line([-0,-0.55,0],[-4,-0.55,0],color = PINK).rotate(-0.3).fade(0.6)
+ l14= Line([-0,-0.35,0],[-4,-0.35,0],color = PINK).rotate(-0.3).fade(0.6)
+ l15= Line([-0.,-0.15,0],[-4,-0.15,0],color = PINK).rotate(-0.3).fade(0.6)
+
+ rel_text = TextMobject("$\\nabla f = \\lambda \\nabla g$",color = TEAL).shift([3,3.2,0]).scale(0.5)
+
+ f_text = TextMobject("$\\nabla f$",color = '#800000').shift([1,1,0]).scale(0.5)
+ g_text = TextMobject("$\\nabla g$").shift([1.2,-0.8,0]).scale(0.5)
+
+ p_text= TextMobject("$P$").shift([1.8,2.6,0]).scale(0.5)
+
+ #---- labelling of level curves
+ l1_text = TextMobject("$w=$ 17").rotate(math.radians(180)).scale(0.4).shift(2.7*DOWN+4.36*LEFT)
+ l2_text = TextMobject("$w=$ 16").rotate(math.radians(180)).scale(0.4).shift(2.46*DOWN+4.36*LEFT)
+ l3_text = TextMobject("$w=$ 15").rotate(math.radians(180)).scale(0.4).shift(2.2*DOWN+4.36*LEFT)
+ l4_text = TextMobject("$w=$ 14").rotate(math.radians(180)).scale(0.4).shift(1.97*DOWN+4.36*LEFT)
+ l5_text = TextMobject("$w=$ 13").rotate(math.radians(180)).scale(0.4).shift(1.74*DOWN+4.36*LEFT)
+ l6_text = TextMobject("$w=$ 12").rotate(math.radians(180)).scale(0.4).shift(1.5*DOWN+4.36*LEFT)
+ l7_text = TextMobject("$w=$ 11").rotate(math.radians(180)).scale(0.4).shift(1.26*DOWN+4.36*LEFT)
+ l8_text = TextMobject("$w=$ 10").rotate(math.radians(180)).scale(0.4).shift(1.05*DOWN+4.36*LEFT)
+ l9_text = TextMobject("$w=$ 9").rotate(math.radians(180)).scale(0.4).shift(0.8*DOWN+4.32*LEFT)
+ l10_text = TextMobject("$w=$ 8").rotate(math.radians(180)).scale(0.4).shift(0.6*DOWN+4.32*LEFT)
+ l11_text = TextMobject("$w=$ 7").rotate(math.radians(180)).scale(0.4).shift(0.4*DOWN+4.32*LEFT)
+ l12_text = TextMobject("$w=$ 6").rotate(math.radians(180)).scale(0.4).shift(0.2*DOWN+4.32*LEFT)
+ l13_text = TextMobject("$w=$ 5").rotate(math.radians(180)).scale(0.4).shift(-0.02*DOWN+4.32*LEFT)
+ l14_text = TextMobject("$w=$ 4").rotate(math.radians(180)).scale(0.4).shift(-0.23*DOWN+4.32*LEFT)
+ l15_text = TextMobject("$w=$ 3").rotate(math.radians(180)).scale(0.4).shift(-0.44*DOWN+4.32*LEFT)
+
+ level_Curve = VGroup(l1,l1_text,l2,l2_text,l3,l3_text,l4,l4_text,l5,l5_text,l6,l6_text,l7,l7_text,l8,l8_text,l9,l9_text,l10,l10_text,l11,l11_text,l12,l12_text,l13,l13_text,l14,l14_text,l15,l15_text)
+
+ self.set_camera_orientation(phi=0 * DEGREES, theta = 90*DEGREES)
+ self.add(axes)
+ self.add(label_x)
+ self.add(label_y)
+ self.play(Write(surface))
+ self.wait(1)
+ self.play(ShowCreation(level_Curve))
+ self.wait(1)
+ self.play(ShowCreation(a_df),ShowCreation(a_dg),Write(d))
+ self.wait(1)
+ self.add_fixed_in_frame_mobjects(rel_text)
+ self.add_fixed_in_frame_mobjects(p_text)
+ self.wait(1)
+ self.play(Write(qd))
+ self.wait(1)
+ self.play(ShowCreation(b_df))
+ self.add_fixed_in_frame_mobjects(f_text)
+ self.wait(1)
+ self.play(ShowCreation(b_dg))
+ self.add_fixed_in_frame_mobjects(g_text)
+ self.wait(1)
diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.gif
new file mode 100644
index 0000000..9602283
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.gif
Binary files differ
diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.py
new file mode 100644
index 0000000..bf75dd8
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Lagrange-Multipliers/file3_Optimizing_function_w.r.t_one_constraint.py
@@ -0,0 +1,29 @@
+from manimlib.imports import*
+
+class firstScene(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+ cylinder = ParametricSurface(
+ lambda u, v: np.array([
+ np.cos(TAU * v),
+ np.sin(TAU * v),
+ 2 * (u)
+ ]),checkerboard_colors=[YELLOW_C,YELLOW_D,YELLOW_E]
+ ).fade(0.4) #Resolution of the surfaces
+
+ plane = ParametricSurface(
+ lambda u, v: np.array([
+ u,
+ v,
+ u+v
+ ]),checkerboard_colors=[TEAL_C,TEAL_D,TEAL_E]
+ ).scale(2.5)
+ self.add(axes)
+ self.set_camera_orientation(phi=75*DEGREES,theta=45*DEGREES)
+ self.play(Write(cylinder))
+ self.play(Write(plane))
+ self.wait(1)
+ self.begin_ambient_camera_rotation(rate=0.7)
+ self.wait(5)
+ self.move_camera(phi=35*DEGREES,theta=-45*DEGREES)
+ self.wait(2)