Solving Multiple Integrals: Hints for Proving Identity

[Benny]

Hi, I posted a question some time ago and the suggestion was to use some form of the product rule but I still can't figure out what to do.

Q. Let f(x,y,z) and g(x,y,z) be C^2 scalar functions. Let D be an elementary region in space and $$\partial D$$ be the closed surface that bounds D. Prove that

$$\int\limits_{}^{} {\int\limits_{}^{} {\int\limits_D^{} {\nabla f \bullet \nabla g} dV = \int\limits_{}^{} {\int\limits_{\partial D}^{} {f\nabla g \bullet dS} - \int\limits_{}^{} {\int\limits_{}^{} {\int\limits_D^{} {f\nabla ^2 gdV} } } } } }$$

Can someone give me a hint as to where to start, like any relevant identies which could be of use? Any help is appreciated thanks.

 

[StatusX]

The following identity should help:

$$\nabla \cdot (f \vec A) = \vec A \cdot \nabla f + f \nabla \cdot \vec A$$

Which is just one of the 3D versions of the product rule. (You can substitute $\nabla g$ into this formula)

 

[Benny]

Thanks for the help, I'll try to finish this one off.

Edit: Hmm...I could've sworn that your post made mention of the divergence theorem before...nevermind, I'll keep that in mind.;)

 

[StatusX]

Yea, sorry about that. I had originally told you how to do the problem step by step, but I realized that's not what this forum is for, so I changed it. But yes, the divergence theorem is necessary.

 

[Benny]

Oh ok, it doesn't really matter too much now. The divergence theorem is the only connection I know of between surface and volume integrals so I probably would've used it eventually anyway.