Curl of a field in spherical polar coordinates

[Karacora]

Homework Statement

I have a field w=w_φ(r,θ)e_φ^ (e^ is supposed to be 'e hat', a unit vector)
Find w_φ(r,θ) given the curl is zero and find a potential for w.

Homework Equations

I can't type the matrix for curl in curvilinear, don't even know where to start! I've been given it in the form $\frac{1}{r^2 sin(\theta)}$ multiplied by a determinant with top row:
e_r^, re_θ^, rsin(θ) e_φ^

Second row:
∂/∂r, ∂/∂θ, ∂/∂φ

Third row:
a_r, ra_θ, rsin(θ)a_φ

The Attempt at a Solution

a_r, ra_θ = 0 and my a_φ is w_φ. So I'm left with

$\frac{1}{r^2 sin(\theta)}$ ∂(w_φrsin(θ)e_r^)/∂θ - $\frac{1}{r^2 sin(\theta)}$(re_θ^)∂(w_φrsin(θ))/∂r

So I evaluated the partial derivatives, set each individual component to zero and get a partial differential equation. We have not been taught how to solve these, so I can only assume it's the wrong method... I gave it a go as the two equations are separable but they don't give w as a function of r and θ and I have no initial conditions to find c with!

After differentiating I had:
[rcos(θ)w_φ + rsin(θ) ∂w_φ/∂θ] e_r^ - re_θ^ [sin(θ)w_φ + rsinθ ∂w_φ/∂r] all multiplied by $\frac{1}{r^2 sin(\theta)}$ And both components should individually equal zero, I think.

 

[jedishrfu]

Here are the vector identities in curvilinear coordinates:

http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

There's a version of curl for spherical coordinates listed.

 

[Karacora]

Here are the vector identities in curvilinear coordinates:

http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

There's a version of curl for spherical coordinates listed.

That is an amazing link! Thank you :) It does confirm my matrix is correct...