(a) Find the divergence of the function v =ˆr/ rFirst compute it directly, as in Eq. 1.84. Test your result using the divergence theorem, as in Eq. 1.85. Is there a delta function at the origin, as there was for ˆr/r ^2?What is the general formula for the divergence of r ^n ˆr? [Answer:

Question

(a) Find the divergence of the function

[latex]v=\frac{\hat{r} }{r}[/latex]

First compute it directly, as in Eq. 1.84. Test your result using the divergence theorem, as in Eq. 1.85. Is there a delta function at the origin, as there was for [latex]\hat{r}/r^2 [/latex] ?What is the general formula for the divergence of [latex]r^n \hat{r} [/latex] [Answer: [latex]∇.\left(r^n \hat{r} ight) =(n + 2)r^{n -1} [/latex] , unless [latex]n = −2[/latex], in which case it is [latex]4\pi \delta ^3(r)[/latex] ; for [latex]n < −2[/latex]

, the divergence is ill-defined at the origin.]

(b) Find the curl of [latex]r^n \hat{r} [/latex]. Test your conclusion using Prob. 1.61b. [Answer:

[latex]∇ imes \left(r^n \hat{r} ight)=0 [/latex]]

[latex]∇\cdot v=\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2\frac{1}{r^2} ight) =\frac{1}{r^2} \frac{\partial}{\partial r} (1)=0[/latex] (1.84)

[latex]\oint{}v · da =\int{} \left(\frac{1}{R^2}\hat{r} ight)\cdot \left(R^2\sin heta d heta d\phi\hat{r} ight)=\left(\int_{0}^{\pi }{}\sin heta d heta ight) \left(\int_{0}^{2\pi }{}d\phi ight) =4\pi [/latex] (1.85)

Accepted Answer

(1) [latex]∇\cdot v=\frac{1}{r^2}\frac{\partial}{\partial r} \left(r^2\cdot \frac{1}{r} ight)=\frac{1}{r^2}\frac{\partial}{\partial r} (r)=\frac{1}{r^2}[/latex] .

For a sphere of radius R:

[latex]\left. \begin{matrix}\int{}v\cdot da=\int{}\left(\frac{1}{R}\hat{r} ight)\cdot \left(R^2\sin heta d heta d\phi \hat{r} ight) =R\int{\sin heta d heta d\phi =4\pi R} . \ \int{}\left(∇\cdot v ight)d au =\int{}\left(\frac{1}{r^2} ight) \left(r^2\sin heta drd heta d\phi ight)=\left(\int\limits_{0}^{R}{}dr ight) \left(\int{}\sin heta d heta d\phi ight) =4\pi R . \end{matrix} ight\}[/latex]

So divergence theorem checks.

Evidently there is no delta function at the origin.

[latex]∇ imes \left(r^n \hat{r} ight) =\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2r^\eta ight)=\frac{1}{r^2}\frac{\partial}{\partial r} (r^{n +2} ) = \frac{1}{r^2} (n + 2)r^{n -1} =(n + 2)r^{n -1} [/latex]

(except for [latex]n = −2[/latex], for which we already know (Eq. 1.99) that the divergence is [latex]4\pi \pmb{\delta} ^3(r))[/latex] .

[latex]∇\cdot \left(\frac{\hat{r} }{r^2} ight)=4\pi \delta ^3 (r).[/latex] (1.99)

(2) Geometrically, it should be zero. Likewise, the curl in the spherical coordinates obviously gives zero.
To be certain there is no lurking delta function here, we integrate over a sphere of radius R, using Prob. 1.61(b): If [latex]∇ imes \left(r^n \hat{r} ight)=0 [/latex], then [latex]\int{}\left(∇ imes v ight)d au =0\overset{?}{=}-\oint{}v imes da. [/latex] But [latex]v = r^n \hat{r}[/latex] and [latex]da =R^2\sin heta d heta d\phi \hat{r}[/latex] are both in the [latex] \hat{r} [/latex] directions, so v × da = 0.

Question 1.63: (a) Find the divergence of the function v =ˆr/ rFirst comput...

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