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Math Help - Curl of a Radial field

  1. #1
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    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.
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  2. #2
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    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!!
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  3. #3
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    Gdansk, Poland
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    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
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