nLab Hodge-Maxwell theorem



Differential cohomology


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The Hodge theorem in the language of electromagnetism. Over a Riemann surface this may be regarded as simple case of the Narasimhan-Seshadri theorem.


Let (X,g)(X,g) be a compact oriented Riemannian manifold of dimension nn. Write \star for the corresponding Hodge star operator.

Then for every exact differential n-form jj of degree nk1n-k-1 there is in each de Rham cohomology class of degree k a unique representative closed k-form FF

dF=0 \mathbf{d} F = 0

such that

dF=j. \mathbf{d}\star F = j \,.

Reading this as Maxwell's equations on (X,g)(X,g) then gg is the field of gravity, FF is the Faraday tensor measuring the field strength of the electromagnetic field and jj is the electric current.


The term “Hodge-Maxwell theorem” in the above form appears in

Last revised on February 12, 2024 at 15:25:41. See the history of this page for a list of all contributions to it.