Prove $ \oint_{\partial V} (\mathbf{\hat{n}} \times \mathbf{A}) \; \mathrm{d}S = \int_V (\nabla \times \mathbf{A}) \; \mathrm{d}V   $ I need help proving the following vector calculus identity:  

$$
\oint_{\partial V} (\mathbf{\hat{n}} \times \mathbf{A}) \; \mathrm{d}S = \int_V (\nabla \times \mathbf{A}) \; \mathrm{d}V  
$$  

the identity is also found on this link, under "Surface–volume integrals".

Thank you very much
Since $\mathbf{\hat{n}}$ and $\mathbf{dS}$ are parallel, $\mathbf{dS} \times \mathbf{\hat{n}} = 0$, so
\begin{align*}
\mathbf{\hat{n}} \times (\mathbf{A} \times \mathbf{dS}) &amp; = - \mathbf{A} \times (\mathbf{dS} \times \mathbf{\hat{n}}) - \mathbf{dS} \times (\mathbf{\hat{n}} \times \mathbf{A}) \\
 &amp; = - \mathbf{dS} \times (\mathbf{\hat{n}} \times \mathbf{A}) \\
 &amp; = (\mathbf{\hat{n}} \times \mathbf{A}) \times \mathbf{dS}
\end{align*}
by the the Jacobi Identity and divergence theorem. 
$$\mathbf{A} ~dS = (\mathbf{\hat{n}} \times \mathbf{A}) \times \mathbf{dS} + \mathbf{\hat{n}} \cdot \mathbf{A} ~\mathbf{dS}.$$
Then use the corollaries to show that
$$\iint_{\partial V} \mathbf{A} ~dS = \iiint_V \left(\nabla(\mathbf{\hat{n}} \cdot \mathbf{A})- \nabla \times (\mathbf{\hat{n}} \times \mathbf{A})\right) ~dV.$$
here this guy explains it nicely
http://www.youtube.com/watch?v=1qLb0B40YnA