Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 6 / d

Let $J_{ij}={\int }_B{x}_i{x}_j{d}^{3}x$. Rotational symmetry makes $J$ an isotropic second-rank tensor, so $J_{ij}=C{\delta }_{ij}$. Contracting the indices and integrating in spherical coordinates givesThe isotropic second moment of a ball therefore givesOnly the scalar part of $M$ contributes; its antisymmetric tensor and symmetric-traceless parts give zero.