# Curl of a field in spherical polar coordinates

1. May 5, 2015

### Karacora

1. The problem statement, all variables and given/known data
I have a field w=wφ(r,θ)eφ^ (e^ is supposed to be 'e hat', a unit vector)
Find wφ(r,θ) given the curl is zero and find a potential for w.
2. Relevant 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:
er^, reθ^, rsin(θ) eφ^

Second row:
∂/∂r, ∂/∂θ, ∂/∂φ

Third row:
ar, raθ, rsin(θ)aφ

3. The attempt at a solution
ar, raθ = 0 and my aφ is wφ. So I'm left with

$\frac{1}{r^2 sin(\theta)}$ ∂(wφrsin(θ)er^)/∂θ - $\frac{1}{r^2 sin(\theta)}$(reθ^)∂(wφrsin(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!

[rcos(θ)wφ + rsin(θ) ∂wφ/∂θ] er^ - reθ^ [sin(θ)wφ + rsinθ ∂wφ/∂r] all multiplied by $\frac{1}{r^2 sin(\theta)}$ And both components should individually equal zero, I think.

Last edited: May 5, 2015
2. May 5, 2015

3. May 5, 2015