I recently did an amusing exercise. Starting from
the equations of electromagnetism in the form
del F = 4pi J
I turned the crank on the Clifford Algebra to confirm
that it does indeed regenerate the old-fashioned vector
representation of Maxwell's equations.
It was an interesting tour of the Geometric Algebra
machinery. I wrote up some notes on the scenery
along the way, including some of the pitfalls that
one learns to avoid. http://www.monmouth.com/~jsd/physics/maxwell-ga.htm