215.sagews

# Let's look at the examples on the worksheet first.

var('x,y')
plot_vector_field((sin(x),cos(y)),(x,-pi,pi),(y,-pi,pi))

g = x^2 + x*y
plot_vector_field(g.gradient(),(x,-2,2),(y,-2,2))

# Here's the same plot, except we show some of the level curves of g on the same axes.
# Note how all the gradient vectors are perpendicular to the level curves.

plot_vector_field(g.gradient(),(x,-2,2),(y,-2,2)) +\
    contour_plot(g,(x,-2,2),(y,-2,2),fill=False,plot_points=100,contours=20,cmap='winter')

# Another example similar to the above, with sunnier contours.

var('x,y')
h = e^x*cos(y)
plot_vector_field(h.gradient(),(x,-1,0.5),(y,0,2*pi)) +\
    contour_plot(h,(x,-1,0.5),(y,0,2*pi),fill=False,plot_points=100,contours=15,cmap='autumn')