Let \(\hat{F} = r^n \,\hat{r}\)

\(= r^n (x\hat{i} + y\hat{j} + z\hat{k})\)

\(=x r^n \hat{i} + y r^n\hat{j} + z r^n\hat{k}\)

\(\vec{\Delta} \times \vec{F}\) \(= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ \frac{\delta}{\delta x} & \frac{\delta}{\delta y} & \frac{\delta}{\delta z}\\ xr^n & yr^n & zr^n \end{vmatrix}\)

\(= \hat{i}(nz\,r^{n-1} \frac{\delta r}{\delta y} - ny\,r^{n-1}\frac{\delta r}{\delta z})\) \(+ \hat{j}(nx\,r^{n-1} \frac{\delta r}{\delta z} - nz\,r^{n-1}\frac{\delta r}{\delta x})\) \(+ \hat{k}(ny\,r^{n-1} \frac{\delta r}{\delta x} - nx\,r^{n-1}\frac{\delta r}{\delta y})\)

\(= \hat{i}(nyz\,r^{n - 2} - nyz\,r^{n - 2})\) \( + \hat{j}(nxz\,r^{n - 2} - nxz\,r^{n - 2})\) \( + \hat{k}(nxy\,r^{n - 2} - nxy\,r^{n - 2})\)

\((\because r^2 = x^2 + y^2 + z^2\) \(\Rightarrow \frac{\delta r}{\delta x} = \frac{x}{r},\) \(\frac{\delta r}{\delta y} = \frac{y}{r}\) \(\&\) \(\frac{\delta r}{\delta z} = \frac{z}{r})\)

\(= 0\hat{i} + 0\hat{j} + 0\hat{k}\)

\(= \hat{0}\)

\(\therefore \) For all values of n, vector F is irrational.

Now \(\hat{\Delta}.\hat{F}\) \(= \left(\frac{\delta}{\delta x}\hat{i} + \frac{\delta}{\delta y}\hat{j} + \frac{\delta}{\delta z}\hat{k}\right)\) \(.\left(xr^n\hat{i} + yr^n\hat{j} + zr^n\hat{k}\right)\)

\(= \frac{\delta}{\delta x}xr^n + \frac{\delta}{\delta y}yr^n + \frac{\delta}{\delta z}zr^n\)

\(= (nx\,r^{n-1} \frac{\delta r}{\delta x} + r^n)\) \(+ (ny\,r^{n-1}\frac{\delta r}{\delta y} + r^n)\) \(+ (nz\,r^{n-1}\frac{\delta r}{\delta z} + r^n)\)

\(= 3r^n + (nx^2 r^{n - 2} + ny^2 r^{n - 2}\) \(+ nz^2 r^{n - 2})\)

\((\because \frac{\delta r}{\delta x} = \frac{x}{r}, \frac{\delta r}{\delta y} = \frac{y}{r}\) \(\&\) \(\frac{\delta r}{\delta z} = \frac{z}{r})\)

\(= 3r^n +nr^{n - 2}(x^2 + y^2 + z^2)\)

\(= 3r^n +nr^{n}\) \((\because x^2 + y^2 + z^2 = r^2)\)

\(= (3 + n)r^n\)

\(\therefore \vec{\Delta}.\vec{F} = 0\) when n = -3.

\(\therefore \vec{F} = r^n\vec{r}\) is solenoidal only if n = -3.