vector derivative of r^n

1. gradient

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

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

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

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

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

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

 

2. divergence

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

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

 

3. curl

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

 

4. laplacian

$$\begin{array}{rcl}{\mathrm{\nabla }}^2(r^n)& =& \mathrm{\nabla }\cdot \mathrm{\nabla }(r^n)=\mathrm{\nabla }\cdot (n{r}^{n-1}\hat{r})=n\mathrm{\nabla }\cdot (r^{n-1}\hat{r})\\ & =& \{\begin{array}{ll}n(n+1)r^{n-1}& (n\ne -1)\\ -4\pi \delta (\overrightarrow{r})& (n=-1)\end{array}\end{array}$$