Friday, March 1, 2019

electromagnetism - Maxwell's Equations using Differential Forms


Maxwell's Equations written with usual vector calculus are



E=ρ/ϵ0B=0 ×E=Bt×B=μ0j+1c2Et


now, if we are to translate into differential forms we notice something: from the first two equations, it seems that E and B should be 2-forms. The reason is simple: we are taking divergence, and divergence of a vector field is equivalent to the exterior derivative of a 2-form, so this is the first point.


The second two equations, though, suggests E and B should be 1-forms, because we are taking curl. Thinking of integrals, the first two we integrate over surfaces, so the integrands should be 2-forms and the second two we integrate over paths and so the integrands should be 1-forms.


In that case, how do we represent E and B with differential forms, if in each equation they should be a different kind of form?




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