var('x y t l')
f(x, y) = x^2 + 2 * y^2
g(x, y) = x^2 + y^2
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def gradient(f, X):
return vector([ diff(f, x) for x in X])
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S = solve([diff(f(x, y), x) == l * diff(g(x, y), x),
diff(f(x, y), y) == l * diff(g(x, y), y),
g(x, y) == 1], [x, y, l], solution_dict = True)
S
[{y: 0, x: 1, l: 1}, {y: 0, x: -1, l: 1}, {y: -1, x: 0, l: 2}, {y: 1, x:
0, l: 2}]
[{y: 0, x: 1, l: 1}, {y: 0, x: -1, l: 1}, {y: -1, x: 0, l: 2}, {y: 1, x: 0, l: 2}]
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P = contour_plot(f(x, y), (x, -2.5, 2.5), (y, -1.5, 1.5),
contours = srange(0, 3.01, 0.5), plot_points = 1000, fill = False)
P += parametric_plot([cos(t), sin(t)], 0, 2 * pi, color = 'red')
for s in S:
p = vector([x, y])(s)
P += arrow(p, p + gradient(f(x, y), [x, y])(s), rgbcolor = 'black')
P += arrow(p, p + gradient(g(x, y), [x, y])(s), rgbcolor = 'red')
P += point(p, pointsize = 30)
show(P)
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P = plot3d(f(x, y), (x, -1.5, 1.5), (y, -1.5, 1.5))
P += parametric_plot3d([cos(t), sin(t), 0], (t, 0, 2 * pi), thickness = 2, color = 'red')
P += parametric_plot3d([cos(t), sin(t), f(cos(t), sin(t))], (t, 0, 2 * pi),
thickness = 2, color = 'green')
for s in S:
p = [x(s), y(s), f(x(s), y(s))]
P += point3d(p, size = 10, color = 'black')
show(P)
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var('u v')
P = parametric_plot3d([u * cos(v), u * sin(v), f(u * cos(v), u * sin(v))],
(u, 0, 1), (v, 0, 2 * pi))
P += parametric_plot3d([cos(t), sin(t), 0], (t, 0, 2 * pi), thickness = 2, color = 'red')
P += parametric_plot3d([cos(t), sin(t), f(cos(t), sin(t))], (t, 0, 2 * pi), thickness = 2, color = 'green')
for s in S:
p = [x(s), y(s), f(x(s), y(s))]
P += point3d(p, size = 10, color = 'black')
show(P)
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