Introduction to Electrodynamics · Problem 1.36

Vector Analysis — David J. Griffiths

Physics · vector-calculus-integration-by-parts

Problem

(a) Show that

\int_{\mathcal{S}} f(\nabla \times \mathbf{A}) \cdot d\mathbf{a} = \int_{\mathcal{S}} [\mathbf{A} \times (\nabla f)] \cdot d\mathbf{a} + \oint_{\mathcal{P}} f\,\mathbf{A} \cdot d\mathbf{l}.

(b) Show that

\int_{\mathcal{V}} \mathbf{B} \cdot (\nabla \times \mathbf{A})\,d\tau = \int_{\mathcal{V}} \mathbf{A} \cdot (\nabla \times \mathbf{B})\,d\tau + \oint_{\mathcal{S}} (\mathbf{A} \times \mathbf{B}) \cdot d\mathbf{a}.