How is Hodge theory of harmonic forms related to maxwell's equations.Atiyah says that Hodge was directly motivated by considerations of maxwell's equations while commenting on donaldson.
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In the absence of electric current, Maxwell's equations say precisely that the electromagnetic potential is a harmonic 1form; see Wikipedia. This is in a spacetime manifold, so it isn't the usual Hodge theory. Being harmonic in spacetime is a wave equation, not an electrostatics equation. But if your spacetime is a product of a Riemannian space manifold and a time direction, then time invariance of the electromagnetic potential reduces the equations of motion to harmonicity in the sense of the Hodge Laplacian. 

