Decoding Leibniz Notation (2024)

I remember reading Einstein's Relativity and having to translate the notation into what I was learning in Calculus class.

There's a lot of online discussion that insists Leibniz notation is just Lagrange notation with wacky conventions. This post seems to have the same conclusion, saying that df/dx should be a syntax error. I understand where this is coming from. For example, in the chain rule

dz/dx = dz/dy dy/dx,

the expression dz/dy needs to become z'(y(x)) in Lagrange notation, whereas dy/dx is just y'(x). So it's not possible to map Leibniz notation unambiguously onto Lagrange notation and it's reasonable to conclude there are "vagaries" at play.

However, I promise that this notation is totally vagary-free if you think of the problem of differentiating a _relationship_ between three variables (x, y, z). Specifically, we need to think about how a relationship between these variables "extends" to a relationship between (x, y, z, dx, dy, dz), where (dx, dy, dz) measures an infinitesimal perturbation. In differential geometry we call that operation a "prolongation." Then it makes perfect sense to write an equation like

dz/dx = 4 y.

That just means that, for any infinitesimal nudge we can make to (x, y, z) while respecting our relationship, z gets moved at 4 y times the rate that we moved x. We could also write dz = 4 y dx. If our relationship has one degree of freedom, it is then easy to show that dz/dx = dz/dy dy/dx at any point where all these fractions are well-defined. It's also true that, when our relationship has two degrees of freedom and we modify the "fractions" dz/dx to be such that dz = dz/dx dx + dz/dy dy, then

dz/dx = -dz/dy dy/dx,

which is a formula that can't be copied into Lagrange notation so far as I know.