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

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- Solution a

Solution

Let the new orthonormal basis satisfy

e_{i}^{'} = R_{ij} e_{j}

. Then an

n

th-rank tensor transforms by

T_{i_{1} \dots i_{n}}^{'} = R_{i_{1} j_{1}} \dots R_{i_{n} j_{n}} T_{j_{1} \dots j_{n}} .

Since

\partial / \partial x_{i}^{'} = 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

a, 0, b δ_{ij}, c ϵ_{ijk} .

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