Divergence of a curl ($\nabla \cdot (\nabla \times \mathbf{A})=0$)

For a twice continuously differentiable vector field in Cartesian coordinates, expand the curl and then its divergence. Every mixed partial derivative occurs twice with opposite signs, proving the identity. In electromagnetism this forces the right side of the Ampère-Maxwell equation to be divergence-free, yielding local charge conservation.