Maxwell's Equations written with usual vector calculus are
∇⋅E=ρ/ϵ0∇⋅B=0 ∇×E=−∂B∂t∇×B=μ0j+1c2∂E∂t
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?
No comments:
Post a Comment