2.I.2G

Methods | Part IB, 2001

Show that the symmetric and antisymmetric parts of a second-rank tensor are themselves tensors, and that the decomposition of a tensor into symmetric and antisymmetric parts is unique.

For the tensor AA having components

A=(123456123)A=\left(\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \end{array}\right)

find the scalar aa, vector p\mathbf{p} and symmetric traceless tensor BB such that

Ax=ax+px+BxA \mathbf{x}=a \mathbf{x}+\mathbf{p} \wedge \mathbf{x}+B \mathbf{x}

for every vector x\mathbf{x}.

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