The curl of a vector field: Beyond the formula

It has been widely acknowledged that there is some discrepancy in the teaching of vector calculus in mathematics courses and other applied fields. The curl of a vector field is one topic many students can calculate without understanding its significance. In this paper, we explain the origin of the curl after presenting the standard mathematical formulas. We investigate when and why a vector field yields an in-spot spin, also known as curl, and develop intuition to predict the sign of the curl of a vector field without calculating it. As an application of the curl, Stokes' theorem and its physical interpretation are presented with simple illustrations.