Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 3 / 4A / Solution

Use suffix notation, with repeated indices summed from one to three. The cross product, divergence, and curl have components $(\mathbf{A}\times \mathbf{B}{)}_i={ϵ}_{ijk}{A}_j{B}_k$, $\nabla \cdot \mathbf{A}={\partial }_i{A}_i$, and $(\nabla \times \mathbf{A}{)}_i={ϵ}_{ijk}{\partial }_j{A}_k$. Here $ϵ$ is the Levi-Civita symbol.

The product rule givesUsing the cyclic identity ${ϵ}_{ijk}={ϵ}_{kij}$ in the first term and ${ϵ}_{ijk}=-{ϵ}_{jik}$ in the second identifies these terms as $B_k(\nabla \times \mathbf{A}{)}_k$ and $-A_j(\nabla \times \mathbf{B}{)}_j$. ThereforeFor the second identity, contract two Levi-Civita symbols, using ${ϵ}_{ijk}{ϵ}_{klm}={\delta }_{il}{\delta }_{jm}-{\delta }_{im}{\delta }_{jl}$:The four component expressions translate intoBoth derivations require only continuously differentiable vector fields; no additional divergence-free assumptions are used.