﻿What really is the chain rule?


It basically says that the derivative of a composite function (fg)’ is the product of those functions derivatives: f’g’


Since the domain of f’ is points that g outputs, it makes sense for the full chain rule to be:


f’(g(x))g’(x)


Now in the multidimensional case, in the tangent vector case we have:


(f*x^-1*x*y)’


The way to interpret this function is that:
F: M -> R
x^-1: R^n -> M
X: M -> R^n
Y: R -> M


And the way you should interpret function composition is:


Starting on the right, all of these things will be inputs to the next thing on the left.


Because the most rightword thing is y, that means that the big daddy function takes in an R.


Then that R becomes an M, and becomes the input to the next step x, which becomes a R^n, which becomes the input to the next step x^-1, which becomes M, which becomes the input for the next step f, which becomes R.


So the total function is R->R


In fact the simplified version of that very function, without the local charts, is also a function from R to R.


f*y


Because y takes in an R, which becomes an M, which is the input for f, which becomes an R.


So R -> R.


In any case, in the chart case, we take the chain rule of the first pair and second pair, which are functions R^n -> R, and R -> R^n respectively.


What it actually means to take the chain rule of these multidimensional functions, I still don’t fully understand.