Two ways of approaching a problem can be computationally equivalent and yet one can be a clear winner. For example, take a look at Maxwell’s equations—as originally formulated there were 20 different equations, with many equations duplicated for x, y, and z. Heaviside, Gibbs, and Hertz reformulated these using vectors and reduced the number of equations to four. Computationally equivalent, but nobody uses the 20-equation version of Maxwell’s equations any more.

Now consider relativity, which was originally motivated by Maxwell’s equations. We can describe flat spacetime as a 4D space with the Minkowski metric, which we can extend into a geometric algebra Cl_1,3(R). We can then describe field and current as multivectors, and we end up with the equation (singular!)

∇ F = µ_0 c J

This has the additional property that it captures how observations of electromagnetism change under Lorentz transforms. If you think of a Lorentz transform as just a change of basis in spacetime algebra, and if you think of electricity and magnetism as together being the basis for electromagnetism, then it’s obvious that (for example) a moving observer would see a magnetic field generated by a stationary charge.

This is obvious because the basis change from a stationary to moving observer will directly correspond to a basis change from an electric field to a magnetic field. This is simplifying a bit but I find it easier to remember and reason about the geometric algebra version of many of these formulas.