16-5 Curl and Divergence.sagews

var('x', 'y', 'z')
V = vector([cos(x*y), sin(y)])
V.div([x, y])
G1 = plot_vector_field(V, (x, -pi, pi), (y, -pi, pi))
G2 = implicit_plot(V.div([x, y]) == 0, (x, -pi, pi), (y, -pi, pi))
G3 = contour_plot(V.div([x, y]), (x, -pi, pi), (y, -pi, pi), contours = [-5, 0, 5], cmap = ([1, 0, 0], [0, 1, 0])) # green is positive/red is negative
show(G3 + G1, figsize = 10)

(x, y, z) -y*sin(x*y) + cos(y)

V = vector([cos(x*y), sin(y), 0])
V.curl([x, y, z])
G1 = plot_vector_field((V[0], V[1]), (x, -pi, pi), (y, -pi, pi))
G2 = implicit_plot(V.curl([x, y, z])[2] == 0, (x, -pi, pi), (y, -pi, pi))
G3 = contour_plot(V.curl([x, y, z])[2], (x, -pi, pi), (y, -pi, pi), contours = [-5, 0, 5], cmap=([1,0,0],[0,1,0]))    # green is positive (upward)/red is negative (downward)
show(G3 + G1, figsize = 10)

(0, 0, x*sin(x*y))

var('x', 'y', 'z')
V = vector([y, sin(x), 0])
V.curl([x, y, z])           # note that this is never positive
plot_vector_field((V[0], V[1]), (x, -2*pi, 2*pi), (y, -pi, pi), aspect_ratio = 1, figsize =8)

(x, y, z) (0, 0, cos(x) - 1)