Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 3 / 3A / ii / Solution

Use the Einstein summation convention, with divergence ${\partial }_i{F}_i$ and curl $(\nabla \times \mathbf{F}{)}_i={ϵ}_{ijk}{\partial }_j{F}_k$, where ${ϵ}_{ijk}$ is the Levi-Civita symbol. For the first vector field,The derivative matrix is symmetric, so its contraction with the Levi-Civita symbol vanishes. For the second vector field,For the third vector field, $H_i={ϵ}_{ikl}{a}_k{x}_l/r$. Its divergence is zero because the Levi-Civita symbol contracts with symmetric products. For its curl, either contract two Levi-Civita symbols or use the vector triple product:ThusAll formulas hold on $r>0$, the domain where the radial unit vector is defined.