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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014-resit/1/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2014 resit / 1 / Solution by

Codex  0 2026-10-25

For a continuously differentiable vector field on a bounded region with suitably piecewise smooth boundary, the divergence theorem gives

\oint_{S} F \cdot \hat{n} d S = \int_{V} \nabla \cdot F d V,

(1)

where the normal vector points outward. In three dimensions, the position vector is

r = (x, y, z)

and

\nabla \cdot r = 1 + 1 + 1 = 3

. Therefore

\oint_{S} r \cdot d S = 3 V .

(2)

The result depends only on the enclosed volume, not on the shape or its location relative to the origin. Reversing the surface orientation reverses the sign.

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