 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/1/ii/solution

The divergence theorem states that a continuously differentiable vector field

F

V

with piecewise smooth closed boundary

S

satisfies

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

where

\hat{n}

is the outward unit normal. Take

F = \nabla a \times B

. Since the curl of a gradient is zero for a twice continuously differentiable scalar field, part (i) gives

\nabla \cdot (\nabla a \times B) = - \nabla a \cdot (\nabla \times B) .

Integrating this identity and applying the divergence theorem proves

\int_{V} \nabla a \cdot (\nabla \times B) d V = - \int_{S} (\nabla a \times B) \cdot \hat{n} d S .

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