General form of Integration by Parts

This is a question just out of interest to know the power of integration by parts. There are various level of integration by parts. What are some of the most general form of integration by parts? I have encountered it very often in PDE's. I look forward to gaining more insights on it. Thank you for your ideas, help and discussions.

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To get something new, it wouldn't hurt to see what you already have: Drop your pants! (not literally!!!) – draks ... Jul 17 '12 at 13:06
@draks : I have tried to drop :D. – Theorem Jul 17 '12 at 13:11

All versions of integration by parts that I have seen boil down to two things.

1. Stokes' Theorem: if $\omega$ is an $n-1$ form on $M$, and $n$-manifold with boundary $\partial M$, then $$\int_M \mathrm{d}\omega = \int_{\partial M} \omega$$

2. Leibniz rule for differential forms: $$\mathrm{d}(\eta \wedge \omega) = \mathrm{d}\eta \wedge \omega + (-1)^{\text{degree}(\eta)}\eta\wedge \mathrm{d}\omega$$

The only other ingredient that is sometimes needed is some basic housecleaning coming from Riemannian and/or differential geometry: things like how covariant or coordinate partial derivatives relate to the exterior derivative, and how to write the divergence of a vector field as equivalently the exterior derivative of its dual $n-1$ form.

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Sir , I don't know differential geometry . But i would like to prove that $\int_\Omega \frac{\partial u}{\partial x_i}.v dx=-\int_\Omega\frac{\partial v}{\partial x_i}.u+\int_{\partial \Omega}u.v.neu_i d\sigma$ can you give me some idea . It would also be nice to show this relation in the way you have described. – Theorem Oct 30 '12 at 17:20
@Theorem: let $\eta$ be the $n-1$ form given by $\mathrm{d}x_1 \wedge \mathrm{d}x_2 \wedge \cdots \mathrm{d}x_{i-1} \wedge \mathrm{d} x_{i+1} \wedge \cdots \wedge \mathrm{d}x_n$ (the volume form with the $x_i$ component removed). Let $\omega = u v \eta$. Then $\mathrm{d}\omega = \left( \partial_i u v + u \partial_i v\right) \mathrm{d}x_i \wedge \eta$. Apply Stokes' theorem. – Willie Wong Oct 31 '12 at 8:06