Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 3 / 4A / Solution

Use suffix notation, summing repeated indices, and write ${\partial }_i=\partial /\partial x_i$. For a twice continuously differentiable vector field, mixed derivatives commute. Thus the divergence of its curl satisfiesbecause the derivatives are symmetric in $i,j$, while the Levi-Civita symbol is antisymmetric in those indices. HenceFor the second identity, use the contraction of two Levi-Civita symbols:Rearranging each component proves the vorticity cross-product identityThe second identity needs first derivatives; the first additionally uses equality of the mixed second derivatives.