# Integration by parts Green's formula

I would like to do some kind of integration by parts to $$\int{f(\nabla g \cdot \nabla h)}$$
We know Green's identity holds with $f \equiv 1$. Is there a nice expression with general $f$? Let's say everything here is smooth so the integrals and derivatives make sense.

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downvote... why? – Euler....IS_ALIVE Sep 12 '13 at 2:57

One might note that $\nabla g \cdot \nabla h = \nabla \cdot (g \nabla h) - g \Delta h$. Not sure where to go from here, but this might be a start.
I'm dealing with functions that are $0$ on the boundary anyways, so do green's identity in vector form (choose $f\nabla g$ and $\nabla h$), then apply your identity. This moves derivatives off of the function I wanted to get them off of. – Euler....IS_ALIVE Sep 12 '13 at 2:59
You need $f \nabla g$ to be the gradient of some function, right? Of what function is it the gradient? (I'm just learning this stuff, too...) – Eric Auld Sep 12 '13 at 9:00
I'm using $\int_{U} (\psi \nabla \cdot \bf{\Gamma} + \bf{\Gamma}\cdot \nabla \psi) dV = \oint_{\partial U}{\psi \bf{(\Gamma \cdot n})}ds$ – Euler....IS_ALIVE Sep 12 '13 at 16:44