Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 12B / ii

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- Solution ii

Solution

A tensor is totally antisymmetric tensor if interchanging any two indices changes its sign. Consequently every component with repeated indices vanishes. In

R^{3}

:

for $n = 2$ , the general tensor is $A_{ij} = a_{k} ϵ_{kij}$ , with three free components;
for $n = 3$ , it is $A_{ijk} = a ϵ_{ijk}$ ;
for $n > 3$ , every component repeats an index, so the tensor is zero.

These are all possibilities.

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