**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.