# Math Help - Curl of a Radial field

1. ## Curl of a Radial field

Let f : R3 to R3 be a radial field, i.e. f (x) = g(||x||) x/||x|| for x ≠0. Show that
Curl f = 0
(i)By direct computation
(ii)By using spherical coordinates

thank you.

2. ## Curl of a radial field

Ok, I have managed to show (i) by direct computation; by just plugging in the partial derivatives and it pops out. I've tried to do (ii) but it's not equal to zero............HELP!!

3. Curl in orthogonal curvilinear 3D coordinates is:
$\nabla \times \vec{f} = \frac{1}{h_1h_2h_3} \sum_{ijk} h_i \vec{e}_i \varepsilon_{ijk} \frac{\partial}{\partial x_j} h_k f_k$
Where $h_i$ are Lamé coefficients and $\varepsilon_{ijk}$ is Levi-Civita symbol.

In spherical coordinates $h_r = 1$, $h_\theta = r$, $h_\phi = r\sin\theta$.

Since in our case $\vec{f}$ has only r component $\nabla \times \vec{f}$ can have only $\theta$ and $\phi$ components.

We can see that
$h_rh_\theta h_\phi \bigl(\nabla \times \vec{f}\bigr)_\theta = \frac{\partial}{\partial \phi} h_r f_r - \frac{\partial}{\partial r} h_\phi f_\phi = 0$
because $h_r$ doesn't depend on $\phi$ and $f_\phi = 0$

The same argument for $\phi$ component:
$h_rh_\theta h_\phi \bigl(\nabla \times \vec{f}\bigr)_\phi = \frac{\partial}{\partial r} h_\theta f_\theta - \frac{\partial}{\partial \theta} h_r f_r = 0$