Introduction to differential forms

Two of Maxwell’s equations
∇ · B = 0, ∇ × E = − ∂B
∂t
can be expressed very succintly in this language as dF = 0. The analogue of
theorem 9.2 holds for R n , and shows that
F = d(A 1 dx + A 2 dy + A 3 dy + A 4 dt)
for some 1-form called the potential. Thus we’ve reduced the 6 quantites to just
4.
18 Further reading
For more information about differential forms, see the books “Differential forms
and applications to the physical sciences” by H. Flanders and “Calculus on
manifolds” by M. Spivak. All the physics background can be found in “Lectures
on Physics” by R. Feynman et. al.
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