Here is a question that I've had for $\geq$ 30 years but only now am in a position to ask properly. Electromagnetic waves are "transverse" which according to the textbooks means that the amplitude (in this case E and B fields) are perpendicular to the direction of propagation.
Now, if we have a point source then the amplitude vectors should furnish a vector field defined on a small ball around that source but according to the aforementioned hairy-ball theorem such cannot exist in a continuous fashion. What is the way out of this?
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