- #1

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## Homework Statement

With ##\vec{r}## the position vector and ##r## its norm, we define

$$ \vec{f} = \frac{\vec{r}}{r^n}.$$

Show that

$$ \nabla^2\vec{f} = n(n-3)\frac{\vec{r}}{r^{n+2}}.$$

## Homework Equations

Basic rules of calculus.

## The Attempt at a Solution

From the definition of the Laplacian

$$

\nabla^2 \vec{f} = \nabla \cdot \nabla \vec{f}\\

= \left( \partial_i \partial_j \delta_{ij} \right) \frac{\vec{r}}{r^n}\\

= \vec{r}\left( \partial_i \partial_i \right) r^{-n}\\

= \vec{r}(-n)(-n-1) r^{-n-2}\\

= n(n+1) \frac{\vec{r}}{r^{n+2}}.$$

Clearly this does not agree with the proposed solution. I think my error is assuming ##vec{r}## is constant, but we know this is not always true in other coordinate systems. Any ideas?