How to Verify Electromagnetic Field Equations in MTW Gravitation Problem?

[qinglong.1397]

Homework Statement

Page. 124, MTW Gravitation. Exercise 4.6.

Verify that the expressions given for the electromagnetic field of an oscillating dipole in equations (4.23) and (4.24) satisfy $<b><i>dF</i></b>=0$ everywhere and $<i><b>d*F</b></i>=0$ everywhere except at the origin.

Homework Equations

(4.23) $<i><b>F</b></i>=$real part of $\left\{p_1e^{iω(r-t)}[2\cos θ(1/r^3-iω/r^2)dr\wedge dt+\sin θ(1/r^3-iω/r^2-ω^2/r)rdθ\wedge dt+\sin θ(-iω/r^2-ω^2/r)dr \wedge rdθ]\right\}$

(4.24) $<i><b>*F</b></i>=$real part of $\left\{p_1e^{i(r-t)}[\sin \theta(-i\omega /r^2-\omega^2/r)dt \wedge r\sin\theta d\phi+2\cos\theta (1/r^3-i\omega /r^2)rd\theta \wedge r\sin\theta d\phi+\sin\theta (1/r^3-i\omega /r^2-\omega^2/r)r\sin\theta d\phi\wedge dr]\right\}$

The Attempt at a Solution

It is easy to show that $<b><i>dF</i></b>=0$ and $<i><b>d*F</b></i>=0$ everywhere except at the origin. But how to show $<b><i>dF</i></b>=0$ at the origin, while $<i><b>d*F</b></i>≠0$ at the origin?

 

[member 553083]

I tried to calculate dF and d*F at the origin, but I could not get an expression that is easy to be simplified. Is there any other way to prove this statement?