Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 1 / e

Table of contents
- Solution e

Solution

Apply the product rule to the tensor field, without assuming any symmetry of

Σ

:

\partial_{j} (r_{i} \partial_{k} Σ_{kj}) = δ_{ij} \partial_{k} Σ_{kj} + r_{i} \partial_{j} \partial_{k} Σ_{kj} = \partial_{k} Σ_{ki} + r_{i} \partial_{j} \partial_{k} Σ_{kj} .

Integrating, the divergence theorem converts each of the two divergences into a boundary term:

\int_{V} r_{i} \partial_{j} \partial_{k} Σ_{kj} d V = \int_{S} r_{i} \partial_{k} Σ_{kj} d S_{j} - \int_{S} Σ_{ki} d S_{k} .

The repeated indices are summed according to Einstein summation convention, while

i

labels the component of the resulting vector identity.

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