However, and this is very amusing to me, it turns out that the process of automatic differentiation (see https://en.wikipedia.org/wiki/Automatic_differentiation, the section on dual numbers) works in exactly the same way. Just replace all of the primed symbols (u', v', etc.) with du, dv, etc. and dual numbers are isomorphic to infinitesimals (if I'm using that term correctly.)
However, and this is very amusing to me, it turns out that the process of automatic differentiation (see https://en.wikipedia.org/wiki/Automatic_differentiation, the section on dual numbers) works in exactly the same way. Just replace all of the primed symbols (u', v', etc.) with du, dv, etc. and dual numbers are isomorphic to infinitesimals (if I'm using that term correctly.)