Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2002 / Paper 2 / 6 / a

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Solution

For Cartesian tensors, let

R

be the orthogonal matrix taking old vector components to new ones,

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

. A rank-

n

tensor transforms by one copy of

R

on each index:

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

Repeated indices are summed according to Einstein notation. In general coordinates a contravariant tensor index instead carries the corresponding Jacobian matrix

\partial x^{' i} / \partial x^{j}

, and a covariant tensor index carries its inverse; in orthonormal bases of Cartesian axes their component laws coincide with the displayed form.

An isotropic tensor has identical components after every proper orthogonal transformation, so

T^{'} = T

for every

R

with

R^{T} R = I

and

det R = 1

. Requiring reflection invariance as well is a stronger convention for odd tensor ranks. The Kronecker delta is invariant under all orthogonal transformations, while the Levi-Civita symbol is invariant under proper rotations and acquires a determinant sign under reflections.

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