 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/1/c/solution

\nabla \times a = 0

give

\nabla \cdot (a \times A) = - a \cdot (\nabla \times A)

. On the boundary the scalar triple product gives

(a \times A) \cdot \hat{n} = - a \cdot (\hat{n} \times A)

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

Because

a

is arbitrary, the curl identity follows:

\int_{V} \nabla \times A d V = \int_{S} \hat{n} \times A d S .

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