Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V / 2 / a / ii

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- Solution ii

Solution

Apply the divergence theorem to

F = u \nabla v - v \nabla u

. The product rule gives

\nabla \cdot F = u \nabla^{2} v - v \nabla^{2} u,

because the two gradient inner products cancel. Also

F \cdot n = u \partial_{n} v - v \partial_{n} u

. Thus

∭_{R} (u \nabla^{2} v - v \nabla^{2} u) d V = \iint_{\partial R} (u \partial_{n} v - v \partial_{n} u) d S .

This is Green's second identity, under the usual regularity assumptions permitting these derivatives and boundary integrals.

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