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My professor wrote notes for us because he doesn't really like Jackson's book, but I'm having a very hard time to understand the most basics of his notes. There are a lot of typos and redundancies, but this is not the reason I don't understand. Here there are: http://www.famaf.unc.edu.ar/~reula/Docencia/Electromagnetismo/part1.pdf.

What I understand from the notes (first part of the first chapter): 2 of Maxwell's vacuum equations are equations of evolution. Then it introduces us a theorem that basically states that if I have 2 fields satisfying certain conditions, then there exists a unique solution of Maxwell's evolution equations. I realize that in reality these 2 fields are the electric and magnetic fields.

Now what's the deal with the wave equation? It's only used to prove the theorem? For one moment I've been thinking the wave equation was a solution of Maxwell's equations, but of course it doesn't make any sense, it's an equation... So I thought that the solution to the wave equation was also the solution to Maxwell's evolution equations. Now I realize it's false.

Also, I don't understand the detour into the wave equation taken. What is the purpose of it? I got lost. If someone could explain a bit these things in a few lines (compared to the notes), I'd be grateful. Until now, I didn't even developed an intuition on the material seen yet.