vector derivative of r^n

1. gradient

$$\nabla(r^n) = \nabla\left( (x^2 + y^2 + z^2)^{n/2} \right) $$

$$ \text{x component: } \frac{d}{dx} \left( (x^2 + y^2 + z^2)^{n/2} \right) = \frac{n}{2} (x^2 + y^2 + z^2)^{\frac{n}{2}-1} \cdot 2x = n x (r^2)^{\frac{n}{2}-1} = n x r^{n-2} $$

$$ \Rightarrow (n  x r^{n-2}, n y r^{n-2}, n  z r^{n-2}) = n r^{n-2} (x, y, z) $$

$$  \Rightarrow \nabla(r^n) = n r^{n-2} \vec{r} = n r^{n-1} \hat{r}$$

$$ ( \Rightarrow r^n \hat{r} = \frac{1}{n+1} \nabla(r^{n+1}) ) $$

$$ \begin{eqnarray} n = 1 &:& \nabla(r) = \hat{r} \\ n = 0 &:& \nabla(1) = 0 \\ n = -1 &:& \nabla\left( \frac{1}{r} \right) = -\frac{1}{r^2} \hat{r} \end{eqnarray}$$

 

2. divergence

$$ \nabla \cdot (r^n \hat{r}) = \frac{1}{r^2} \frac{\partial}{\partial r}(r^2 f_r) = \frac{1}{r^2} \frac{\partial}{\partial r}(r^{n+2}) = \frac{n+2}{r^2} r^{n+1} = (n+2) r^{n-1} $$

$$ \begin{eqnarray}\\ n=1 &:& \nabla \cdot (r \hat{r}) = \nabla \cdot \vec{r} = 3 \\n=0 &:& \nabla \cdot \hat{r} = \frac{2}{r} \\n=-1 &:& \nabla \cdot (\frac{\hat{r}}{r}) = \frac{1}{r^2} \\ n=-2 &:& \nabla \cdot (\frac{\hat{r}}{r^2}) = 4\pi \delta(r) \\ \end{eqnarray} $$

 

3. curl

$$ \nabla \times (r^n \hat{r} ) = \nabla \times (\frac{1}{n+1} \nabla (r^{n+1}) ) = 0 \text{ (curl of gradient=0) } $$

 

4. laplacian

$$ \begin{eqnarray} \nabla^2 (r^n) &=& \nabla \cdot \nabla(r^n) = \nabla \cdot (n r^{n-1} \hat{r} ) = n \nabla \cdot (r^{n-1} \hat{r}) \\ &=& \begin{cases} n(n+1)r^{n-1}  & (n \neq -1) \\ -4\pi \delta (\vec{r}) & (n =-1) \end{cases} \end{eqnarray} $$