# surface and partial dervatives PRoof

• Jul 10th 2013, 08:23 PM
n22
surface and partial dervatives PRoof
Hi,
Guidance is appreciated.Thanks.

Prove that ∫∫∫v∇f.∇gdV+∫∫∫vf∇²gdV=∫∫sf∇g.ndS
V is closed bounded region in space ,whose boundary is piecewise smooth orientable surface S.

n is an outwards pointing unit normal vector to S.
f and g are scalar functions with continuous second partial derivatives.

• Jul 11th 2013, 01:21 PM
Phantasma
Re: surface and partial dervatives Proof
This is actually a simple application of the generalized Stokes' Theorem.

All you need to prove is that $d\left(\langle f\nabla g, \hat{n}\rangle dS\right)=\left(\langle\nabla f, \nabla g\rangle + f\nabla^2g\right)dV$, where I've used angle brackets to denote the inner product.