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Differential Forms Integration Exercise

  1. Feb 14, 2010 #1
    Hi all, I posted this awhile back in the homework sections of the forums and received only one reply, which suggested that I post it here instead, though I understand that it belongs in the homework section. The fundamental problem is not this particular exercise but about integration of differential forms in general. But this is best illustrated in an example and, for simplicity, I just copied the original post:

    1. The problem statement, all variables and given/known data

    The problem I am facing is 2.9 in Sean Carroll's book on general relativity (Geometry and Spacetime) I should note that I am not studying this formally and so a full solution would not be unwelcome, though I understand that forum policy understandably prohibits it. The full problem is stated as

    In Minkowski space, suppose that [tex]*F = q \sin{\theta} d\theta\wedge d\phi[/tex].

    a.) Evaluate [tex]d*F=*J[/tex]
    b.) What is the two-form [tex]F[/tex] equal to?
    c.) What are the electric and magnetic fields equal to for this solution?
    d.) Evaluate [tex]\int_V d*F[/tex] where [tex]V[/tex] is a ball of radius [tex]R[/tex] in Euclidean three-space at a fixed moment of time.

    2. Relevant equations

    In the above the asterisk denotes the Hodge dual and the [tex]d[/tex] denotes the exterior derivative. The definitions of these operators should be well known.

    3. The attempt at a solution

    I think I have the first three parts solved. For part a.) I arrived at the lengthy result of

    [tex]
    \frac{1}{2}[\partial_\mu(*F)_{\nu\rho} - \partial_\nu(*F)_{\mu\rho} + \partial_\nu(*F)_{\rho\mu} - \partial_\rho(*F)_{\nu\mu} + \partial_\rho(*F)_{\mu\nu} - \partial_\mu(*F)_{\rho\nu}] = \epsilon^\sigma_{\,\,\mu\nu\rho}J_\sigma
    [/tex]

    For part b.) I got (with the help of my TI-89) the two-form, in matrix form,

    [tex]
    F = -(**F) = \left[
    \begin{array}{cccc}
    0 & \frac{q}{r^4 \sin{\theta}} & 0 & 0\\
    \frac{-q}{r^4 \sin{\theta}} & 0 & 0 &0\\
    0 & 0 & 0 & 0\\
    0 & 0 & 0 & 0
    \end{array}
    \right]
    [/tex]

    and for c.) I arrived at, directly from part b.),

    [tex]E_r = \frac{-q}{r^4 \sin{\theta}}[/tex]
    [tex]E_\theta = 0[/tex]
    [tex]E_\phi = 0[/tex]

    [tex]B_\mu = 0[/tex]

    For [tex]\mu=1,2,3[/tex].

    Verification of these would be appreciated but I am especially confused about the last part d.). In particular the integrand is a three-form and I have no idea how to integrate this and how to transform the integral into a familiar volume integral that can be computed by standard multivariable calculus.
     
  2. jcsd
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