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Math Help - Laplacian of energy potential

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    Laplacian of energy potential

    Hi,

    I have two energy potentials:

    \phi = \int_{V}\frac{1}{|\mathbf{r}-\mathbf{r}'|}dv

    \psi = \int_{V}|\mathbf{r}-\mathbf{r}'|dv

    where \mathbf{r}' is a constant vector.
    Apparently the following is true:

    \nabla^2 \psi = 2\phi
    \nabla^4 \psi = 2\nabla^2\phi

    I did this a few years back and since then my math has become rather rusty but if anyone could explain how I got the first equality I could work on the next one.

    This is my working without including the constant vector:

    Attempt 1:

    \frac{d|\mathbf{r}|}{dx} = \frac{d(\mathbf{r} \cdot \mathbf{r})^{1/2}}{dx} = \frac{\mathbf{r}_{,x}\mathbf{r}}{|\mathbf{r}|}

    where (,x) refers to the partial derivative with respect to x.
    But then this doesn't give the desired result when applying it for all three directions.

    Attempt 2 (Just blindly differentiate with respect to the magnitude r):

    \frac{d|\mathbf{r}|}{dr} = \frac{d(rr)^{1/2}}{dr} = \frac{r}{|r|}

    differentiating again with respect to r:

    \frac{|r|-\frac{r}{|r|}r}{|r|^2}=0


    I've also tried the vector laplacian but it still didn't work.

    Thanks for any help.

    K

    EDIT: I don't know why I did attempt 2 that is just 1 right off the bat.

    Last edited by Krahl; April 13th 2012 at 04:28 AM.
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