Introduction to Electrodynamics · Problem 1.63

Vector Analysis — David J. Griffiths

Physics · divergence-curl-special-functions

Problem

(a) Find the divergence of the function v=r^r.\mathbf{v} = \frac{\hat{\mathbf{r}}}{r}. 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 r^/r2\hat{\mathbf{r}}/r^2? What is the general formula for the divergence of rnr^r^n \hat{\mathbf{r}}? [Answer: ∇⋅(rnr^)=(n+2)rn−1\nabla \cdot (r^n \hat{\mathbf{r}}) = (n+2)r^{n-1}, unless n=−2n = -2, in which case it is 4πδ3(r)4\pi\delta^3(\mathbf{r}); for n<−2n < -2, the divergence is ill-defined at the origin.] (b) Find the curl of rnr^r^n \hat{\mathbf{r}}. Test your conclusion using Prob. 1.61b. [Answer: ∇×(rnr^)=0\nabla \times (r^n \hat{\mathbf{r}}) = \mathbf{0}]

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