Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 3 / 12B / a / Solution

Let the new orthonormal basis satisfy $e_i^{\prime }=R_{ij}{e}_j$. Then an $n$th-rank tensor transforms bySince $\partial /\partial x_i^{\prime }=R_{ij}\partial /\partial x_j$, the Hessian of a scalar obeys this law with two factors of $R$.

An isotropic tensor is unchanged under every rotation. In three dimensions the most general isotropic tensors of ranks zero through three are respectively