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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 3

A second-rank tensor is antisymmetric when
$$
T_{ij}=-T_{ji}.
$$
Under an orthogonal Cartesian coordinate change with matrix $R$,
$$
T'_{ij}=R_{ik}R_{j\ell}T_{k\ell}.
$$
Therefore
$$
\begin{aligned}
T'_{ji}
&=R_{jk}R_{i\ell}T_{k\ell}\\
&=-R_{jk}R_{i\ell}T_{\ell k}\\
&=-T'_{ij},
\end{aligned}
$$
after relabelling the dummy indices. Hence the <antisymmetric second-rank tensor> property is independent of Cartesian coordinates.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 4B / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 4B / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 4B / a / Solution

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