Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 344 / 3 / b / Solution

Put $\mathbf{t}=\mathbf{\theta }$. The product rule applied to $Q_{ij}=\lambda (t_i{t}_j-{\delta }_{ij}/2)$ givesSince $\mathrm{Tr}{Q}^2={\lambda }^2/2$, substituting this vector into the divergence elastic term proves the requested form of the local density. For the radial amplitude and tangential director, the polar coordinates identities areHence the tangential integer nematic defect core hasThe $1/r$ director-curvature term survives when the amplitude is constant. Replacing the divergence norm by a full tensor-gradient norm would not give this expression; it is the PDF's divergence convention that is being evaluated.