 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-v/2/a/iii/solution

Choose

F = \nabla ψ

in the preceding product rule and apply the divergence theorem. Since

\nabla \cdot \nabla ψ = Δ ψ

∭_{V} (ϕ Δ ψ + \nabla ϕ \cdot \nabla ψ), d V = \iint_{\partial V} ϕ \partial_{n} ψ d S .

Here

\partial_{n} ψ = n \cdot \nabla ψ

is the outward normal derivative. Interchange

ϕ

and

ψ

, then subtract. The gradient dot products cancel, yielding the second of Green's identities:

∭_{V} (ϕ Δ ψ - ψ Δ ϕ) d V = \iint_{\partial V} (ϕ \partial_{n} ψ - ψ \partial_{n} ϕ) d S .

For example,

ϕ, ψ \in C^{2} (\overline{V})

with the needed extensions suffice for these applications of the divergence theorem.

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