- #1
member 428835
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?