Calculate the curl of each of the following vector fields. You may look up the formulas for curl in curvilinear coordinates.
$\FF=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z}$
$\GG = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z}$
$\HH = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z}$
$\II = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z}$
$\JJ = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z}$
Compare the curl to the divergence for each field (see Homework 2 Practice).
For each vector field in the preceding problems which have zero curl, find the corresponding potential function.
Choose some simple vector fields of your own and find the curl of them both by hand and using Mathematica or Maple. Choose some that are written in terms of rectangular coordinates and others in cylindrical and/or spherical.
If you need more practice visualizing curl, go through the Mathematica Notebook on the course website.