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: ∇⋅(∇×𝐅)=𝟎\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…