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
Curl in orthogonal curvilinear 3D coordinates is:
Where are Lamé coefficients and is Levi-Civita symbol.
In spherical coordinates , , .
Since in our case has only r component can have only and components.
We can see that
because doesn't depend on and
The same argument for component: