Let $\Omega$ be a bounded domain Lipschitz domain and $A, B\in C^1(\Omega)\cap C(\overline\Omega)$, $u\in C^2(\Omega)\cap C^1(\overline{\Omega})$. Then $ \begin{aligned} \int_{\Omega}\text{curl}\, A\, d\Omega &= \int_{\Gamma} n\times A\, d\Gamma\\ \int_{\Omega}(B\cdot \text{curl}\, A - A\cdot \text{curl}\, B)\,d\Omega &= \int_{\Gamma}(n\times A)\cdot B\,d\Gamma\\ \int_{\Omega}(u\,\text{div}\, A + A\cdot \nabla u)d\Omega &= \int_{\Gamma}u\,(n\times A)\,d\Gamma. \end{aligned} $ The first equality follows from $ \begin{aligned} \int_{\Omega}(\text{curl}\,A)_1\, d\Omega &= \int_{\Omega} \left(\frac{\partial A_3}{\partial x_2} - \frac{\partial A_2}{\partial x_3}\right)\,d\Omega\\ &= \int_{\Omega}\text{div}\, \begin{pmatrix}0\\A_3\\-A_2\end{pmatrix}\,d\Omega\\ &=\int_{\Gamma}n\cdot\begin{pmatrix}0\\A_3\\-A_2\end{pmatrix}\,d\Gamma\\ &=\int_{\partial\Omega}(n\times A)_1\,d\Gamma \end{aligned} $ with the other two components done similarly. The second equation follows from the [[Divergence Theorem]] theorem with $F = A\times B$ and $\text{div}\, F = B\cdot\text{curl}\, A - A \cdot \text{curl}\, B$. The third identity follows again from the divergence theorem with $F=uA$, $\text{div}\, F = u\text{div}\,A + A \cdot\nabla u$, and $n\cdot F = u(n\cdot A)$. # Related Items [[Divergence Theorem]] # Citations A. Kirsch, F. Hettlich, _The Mathematical Theory of Time-Harmonic Maxwell’s Equations,_ Theorem A.13