 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/1/ii/solution

For a sufficiently smooth vector field

A

on a volume with piecewise smooth closed boundary and outward unit normal vector, the divergence theorem states

\int_{V} \nabla \cdot A d V = \int_{S} A \cdot \overset{n}{^} d S .

\nabla \cdot (F \times G) = G \cdot (\nabla \times F) - F \cdot (\nabla \times G)

. Apply the theorem to

A = F \times G

to obtain

\int_{V} [G \cdot (\nabla \times F) - F \cdot (\nabla \times G)] d V = \int_{S} (F \times G) \cdot \overset{n}{^} d S .

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