Does anybody happen to know the integration by parts formula for $\int\varphi\operatorname{curl}(u)dV$, where $u$ is a 3D vector and $\varphi$ is a scalar? Is there a good reference for similar formulae?
Thanks!
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$$\begin{align} \int_V \phi(\vec r)\nabla \times \vec u(\vec r)\,dV&=\int_V \left(\nabla \times(\phi(\vec r)\vec u(\vec r))-\nabla \phi(\vec r)\times \vec u(\vec r)\right)\,dV\\\\ &=\oint_S \phi(\vec r)\left(\hat n \times \vec u(\vec r)\right)\,dS-\int_V \nabla \phi(\vec r)\times \vec u(\vec r)\,dV \end{align}$$
Mark Viola
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it worked perfectly! – Wenbo Zhang Dec 29 '16 at 21:02
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Pleased to hear! And Happy Holidays! -Mark – Mark Viola Dec 29 '16 at 21:08
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I have done so, but was informed that since my score is less than 15 my vote is recorded but not reflected. Sorry, will check later after i am promoted – Wenbo Zhang Dec 29 '16 at 21:25
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Here is a good reference. For your purpose, you need this formula $$\mathop{\rm curl}(\varphi {\bf u})= \varphi \mathop{\rm curl}({\bf u}) + (\mathop{\rm grad} \varphi) \times {\bf u}. $$
After interation, we obtain (employing Stokes' theorem) $$ \int_\Omega \varphi \mathop{\rm curl}({\bf u}) \,dV = \int_{\partial \Omega} \varphi \, \hat{\bf n} \times {\bf u}\, dS - \int_\Omega (\mathop{\rm grad} \varphi) \times {\bf u} \,dV .$$ Note that in typical applications, the surface integral $dS$ vanishes.
A reference for the Stokes' theorem follow this link.
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That is flawed. The surface integral as written is a scalar, while the volume integrals are vectors. – Mark Viola Dec 29 '16 at 20:43
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This isn't really Stokes' Theorem, but rather a disguised Divergence Theorem. ;-)) – Mark Viola Dec 29 '16 at 21:36