Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V resit / Question 2 / a / iii

Table of contents
- Solution iii

Solution

Apply the divergence theorem to

ϕ c

, using ii. Because the vector

c

is constant, it can be taken outside both integrals:

c \cdot ∭_{V} \nabla ϕ d V = ∭_{V} \nabla \cdot (ϕ c) d V = \iint_{\partial V} ϕ c \cdot n d S = c \cdot \iint_{\partial V} ϕ n d S .

This holds for every

c

, in particular each of the three Cartesian basis vectors. Equality of the three components proves the gradient volume-to-boundary identity

∭_{V} \nabla ϕ d V = \iint_{\partial V} ϕ n d S .

It suffices that

ϕ

be a continuously differentiable function on a neighbourhood of the closure of

V

.

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