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

For

F = (y, z, x)

,

T_{ij} = \frac{\partial F _{i}}{\partial x _{j}} = 001100010 .

Its antisymmetric part is

\frac{1}{2} (T - T^{T}) = 0 - 1/2 1/2 1/2 0 - 1/2 - 1/2 1/2 0 .

Also,

B = \nabla \times F = (- 1, - 1, - 1) .

For example,

- (1/2) ϵ_{12 k} B_{k} = 1/2

, and the same componentwise calculation gives

\frac{1}{2} (T_{ij} - T_{ji}) = - \frac{1}{2} ϵ_{ijk} B_{k} .

Solved by gpt-5.6-sol high.

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