Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 3 / a

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Solution

Apply the product rule to the vector field

F = ϕ \nabla ψ - ψ \nabla ϕ

\nabla \cdot F = \nabla ϕ \cdot \nabla ψ + ϕ \nabla^{2} ψ - \nabla ψ \cdot \nabla ϕ - ψ \nabla^{2} ϕ = ϕ \nabla^{2} ψ - ψ \nabla^{2} ϕ .

(1)

The two mixed dot products cancel. The divergence theorem, using the outward unit normal, then gives Green's second identity:

\int_{V} (ϕ \nabla^{2} ψ - ψ \nabla^{2} ϕ) d V = \int_{S} (ϕ n \cdot \nabla ψ - ψ n \cdot \nabla ϕ) d S .

(2)

The usual assumptions are a sufficiently regular boundary and fields with the derivatives and integrability needed for the theorem; singular sources can instead be treated by excising small balls or using distributions.

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 3 / a

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