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Ben Niehoff
#44
Feb23-12, 07:04 PM
Sci Advisor
P: 1,587
Quote Quote by pervect View Post
I think part of the machinery that makes field lines work is that you can write a general two-form as the sum of two wedge products of one-forms in 4 dimensions, and this is peculiair to 4-d.
This has nothing to do with it. Field lines work because the source-free Maxwell equations imply that the 2-form F is harmonic.

Harmonic forms (in any dimension) have the distinction that they capture purely topological information. If you integrate a harmonic n-form over a closed n-surface, the result is either zero or non-zero, depending on whether the n-surface encloses some topological feature (for example, a 1-surface on a cylinder might wrap around the cylinder...or a 2-surface in R^3 might enclose a charge). Any n-surface that encloses the same set of topological features must give you the same result.

You can think of this as a higher-dimensional analogue of contour integration. In fact, all analytic functions on the complex plane satisfy Laplace's equation, which is why contour integration works.