Science by Katharine

If you’ve had multivariable calculus or introductory electromagnetism or fluid mechanics you’ve learned the very useful fact that the divergence of a curl is always zero: $\nabla\cdot\left(\nabla\times {\bf F}\right)={\bf 0}$. Checking this identity is a straightforward calculation, but why does this work?

The identity isn’t a mathematical accident from lucky cancellations: it’s a necessary consequence of Stokes’ Theorem and the Divergence theorem working together. Here’s how to see it in pictures.