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Curl of a field in spherical polar coordinates

  1. May 5, 2015 #1
    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!

    After differentiating I had:
    [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. jcsd
  3. May 5, 2015 #2


    Staff: Mentor

  4. May 5, 2015 #3
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