91Ó°ÊÓ

A vector function that has zero divergence and zero curl everywhere has to be obtained.

The integral of derivative of a function $\mathit{f}\mathbf{}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{z}\mathbf{)}$ over an open surface area is equal to the volume integral of the function, $\mathbf{∫}\mathbf{(}\mathbf{∇}\mathbf{.}\mathbf{v}\mathbf{)}\mathbf{.}\mathbf{d}\mathbf{Ï„}\mathbf{=}{\mathbf{âˆ\circledR }}_{\mathbf{s}}\mathbf{v}\mathbf{-}\mathbf{d}\mathbf{a}$.

The divergence of vector function $F(r,\mathrm{θ},\mathrm{ϕ})$ in spherical coordinates is

$∇F(r,\mathrm{θ},\mathrm{ϕ})=\frac{1}{r^2}\frac{∂\left(r^2{F}_1\right)}{∂r}+\frac{1}{r\sin \mathrm{θ}}\frac{∂\left(\sin \mathrm{θ}{F}_2\right)}{∂\mathrm{θ}}+\frac{1}{r\sin \mathrm{θ}}\frac{∂F_3}{∂\mathrm{ϕ}}$

Here, $r,\mathrm{θ},\mathrm{ϕ}$ are the spherical coordinates.

Let the required function is $\mathrm{v}=\frac{1}{{\mathrm{r}}^2}\hat{\mathrm{r}}$and the del operator is defined as

$∇=\frac{∂}{∂x}i+\frac{∂}{∂y}j+\frac{∂}{∂z}k$. The divergence of vector v is computed as follows:

$$\begin{gathered} ∇.v=\frac{1}{r}\frac{∂\left(r^2{v}_r\right)}{∂r}+\frac{1}{r\sin \mathrm{θ}}\frac{∂\left(\sin \mathrm{θ}{v}_{\mathrm{θ}}\right)}{∂\mathrm{θ}}+\frac{1}{r\sin \mathrm{θ}}\frac{∂\left(v_{\mathrm{φ}}\right)}{∂\mathrm{φ}} \\ =\frac{1}{r}\frac{∂\left(r^2{\displaystyle \frac{1}{r^2}}\right)}{∂r}+\frac{1}{r\sin \mathrm{θ}}\frac{∂\left(\sin \mathrm{θ}\right(0\left)\right)}{∂\mathrm{θ}}+\frac{1}{r\sin \mathrm{θ}}\frac{∂\left(0\right)}{∂\mathrm{φ}} \\ =\frac{1}{r}\frac{∂\left(1\right)}{∂r}+0+0 \\ =0 \end{gathered}$$

The curl of vector v is calculated as follows:

$$\begin{gathered} ∇.v=\left[\begin{array}{cc}\frac{1}{r\sin \mathrm{θ}}& \left[\frac{∂\left(\sin \mathrm{θ}{v}_{\mathrm{θ}}\right)}{∂\mathrm{θ}}-\frac{∂\left(v_{\mathrm{φ}}\right)}{∂\mathrm{φ}}\right]\hat{r}+\frac{1}{r}\left[\frac{1}{\sin \mathrm{θ}}\frac{∂\left(v_r\right)}{∂\mathrm{φ}}-\frac{∂\left(r{v}_{\mathrm{φ}}\right)}{∂r}\right]\widehat{\mathrm{θ}}\\ & +\frac{1}{r}\left[\frac{∂\left(r{v}_{\mathrm{φ}}\right)}{∂r}-\frac{\left(r{v}_r\right)}{∂\mathrm{θ}}\right]\widehat{\mathrm{ϕ}}\end{array}\right] \\ =\left[\begin{array}{cc}\frac{1}{r\sin \mathrm{θ}}& \left[\frac{∂\left(\sin \mathrm{θ}\right(0\left)\right)}{∂\mathrm{θ}}-\frac{∂\left(0\right)}{∂\mathrm{φ}}\right]\hat{r}+\frac{1}{r}\left[\frac{1}{\sin \mathrm{θ}}\frac{∂\left({\displaystyle \frac{1}{r^2}}\right)}{∂\mathrm{φ}}-\frac{∂\left(r\right(0\left)\right)}{∂r}\right]\widehat{\mathrm{θ}}\\ & +\frac{1}{r}\left[\frac{∂\left(r\right(0\left)\right)}{∂r}-\frac{\left(r\left({\displaystyle \frac{1}{r^2}}\right)\right)}{∂\mathrm{θ}}\right]\widehat{\mathrm{ϕ}}\end{array}\right] \\ =0 \end{gathered}$$

Therefore, curl of v is $∇×v=0$.

Hence, for the vector $\mathrm{v}=\frac{1}{{\mathrm{r}}^2}\hat{\mathrm{r}}$the divergence and curl is 0 everywhere