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Math Help - surface and partial dervatives PRoof

  1. #1
    n22
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    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.


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  2. #2
    Newbie Phantasma's Avatar
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    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.
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