\(\hat r=x\hat i+y\hat j+z\hat k\)
\(\therefore\) r2 = x2 + x2 + y2 + z2
(a) \(\vec ▽\).\(\frac{\vec r}{r^3}=\frac1{r^3}\vec ▽.\vec r-\frac3{r^4}(\vec ▽r)\vec r\)
= \(\frac1{r^3}((\frac{\partial}{\partial x}\hat i+\frac{\partial}{\partial y}\hat j+\frac{\partial}{\partial z}\hat k).(x\hat i+y\hat j+\hat k))\)
\(\frac{-3}{r^4}.\frac{\vec r.\vec r}r\)
(\(\because \vec ▽.r= ((\frac{\partial}{\partial x}\hat i+\frac{\partial}{\partial y}\hat j+\frac{\partial}{\partial z}\hat k).\)\(\sqrt{x^2+y^2+z^2}\)
= \(\frac1{2\sqrt{x^2+y^2+z^2}}\)(2x\(\hat i\) + 2y\(\hat j\) + 2z\(\hat k\))
= \(\frac{\vec r}r\))
= \(\frac3{r^3}-\frac3{r^5}.r^2\) (\(\because\) \((\hat r.\hat r = r^2)\))
= \(\frac3{r^3}-\frac3{r^3}\) = 0
(b) \(\vec ▽\times(\frac{\vec r}{r^3})\) = \(\begin{vmatrix}\hat i&\hat j&\hat k\\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ \frac{x}{r^3}&\frac{y}{r^3}&\frac{z}{r^3}\end{vmatrix}\)
\(=\hat i(\frac{\partial}{\partial y}\frac{z}{r^3}-\frac{\partial}{\partial z}\frac y{r^3})\) - \(\hat j(\frac{\partial}{\partial x}\frac{z}{r^3}-\frac{\partial}{\partial z}\frac x{r^3})\) + \(\hat k(\frac{\partial}{\partial x}\frac{y}{r^3}-\frac{\partial}{\partial y}\frac x{r^3})\)
= \(\hat i(-\frac{3zy}{r^4}-(\frac{3yz}{r^4}))-\hat j(-\frac{3xz}{x^4}-(-\frac{3xz}{x^4}))+\hat k(\frac{-3yx}{r^3}-(\frac{-3yx}{r^3}))\)
= 0\(\hat i+\) + 0\(\hat j\) + 0\(\hat k\) = \(\vec 0\)
