Consider in Cartesian coordinates the three-dimensional vector field ?? = u1(x, y, z)?? + u2(x, y, z)?? + u3(x, y, z)??. The curl of ??, denoted as ? × ?? and pronounced “del-cross-u”, is defined as the vector field given by ? × ?? . Here, the cross product is used between a vector differential operator and a vector field. The curl measures how much a vector field rotates, or curls, around a point. A more math-based description will be given later. Example: Show that the curl of a gradient is zero, that is, ? × (?f) = 0. We have ? × (?f) , using the equality of mixed partials. Example: Show that the divergence of a curl is zero, that is, ? · (? × ??) = 0. We have ? · (? × ??) , again using the equality of mixed partials.