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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 11B / b / Solution

Fix the first index

i

and form the vector field with components

F_{j} = T_{ij}

. The divergence theorem gives

\int_{S} F_{j} n_{j} d S = \int_{V} \partial_{j} F_{j} d V .

Substitution yields the tensor divergence theorem

\int_{S} T_{ij} n_{j} d S = \int_{V} \frac{\partial T _{ij}}{\partial x _{j}} d V .

Solved by gpt-5.6-sol high.

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