Computational advantages of various notations for electromagnetism Most undergraduate electromagnetism classes and textbooks use vector notation to describe Maxwell's equations.  However, there are other notations like differential geometry and geometric calculus that simplify the equations and derivations using them.  (See, for example, E&M and geometry - a historical perspective or the paper Teaching electromagnetic field theory using differential forms by Warnick et al.)
My question is this: does using differential forms or geometric calculus give any computational (particularly, paper-and-pencil) advantage in non-relativistic electromagnetism?
In other words, if I'm trying to find the electromagnetic fields and/or potentials of a particular system, will these other notations yield shorter calculations?  Can you please give a specific (nontrivial) example?
 A: Here's a paper for you to ponder on:
Teaching electromagnetic field theory using differential forms
Excerpt from the abstract:

computational simplifications result from the use of forms:
  derivatives are easier to employ in curvilinear coordinates,
  integration becomes more straightforward, and families of vector
  identities are replaced by algebraic rules.

A: There are definite advantages of differential forms, geometric calculus, and four vectors over 3d Gibbs vector algebra and vector calculus.
Specifically you can solve for the electromagnetic field first ... and then let someone break it into electric and magnetic parts later if they feel like it (if at all).
For instance the field due to a non accelerating point charge can get its direction expressed as an outer product of 1) the four velocity of the charge producing the field at the event of the intersection of the past light cone of the event having the field and the world line of the charge producing the field and 2) that null vector connecting the two events. In the instantaneously comoving inertial frame of the charge producing the field back then, it's all electric field. But you don't have to pick a frame and compute an electric field and a magentic field. You can just compute the Faraday field (electromagnetic field) directly.
