 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/1/a/solution

For a bounded volume

V

with piecewise smooth closed boundary

S = \partial V

, and a continuously differentiable vector field

F

on a neighbourhood of

\overline{V}

\iint_{S} F \cdot \hat{n} d S = ∭_{V} \nabla \cdot F d V .

Here

\hat{n}

is the outward unit normal vector. The left side is the outward flux; the right side integrates the divergence over the enclosed volume.

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