TITLE: Chain Rule of Calculus as a Group Property?
QUESTION [5 upvotes]: I read that the chain rule and inverse function theorem are expressions of the group property of successive non-singular transformations. 
How do you say this more formally? 
My guess is that we are saying something like 'the action of a derivation on the algebra of differentiable functions is invariant under the action of the group of non-singular transformations on the algebra'? How would you clean this up and perhaps illustrate with an example?
The quoted passage mentions the implicit function theorem too, I don't see this as directly relating to a group of coordinate transformations (even though it is equivalent to the inverse function theorem), perhaps it's talking about another group property?
Are other theorems from calculus like the mean value theorem or Lagrange multipliers talking about group properties?
For example, Young's theorem seems to be about about the action of one-parameter groups on a function being commutative, though I don't know how to phrase this in a nicer way either.
Also the definition of a convexity seems like it comes from the action of an affine group on a function v.s. it's coordinates, i.e. it's commutator is positive or negative?
Any help? :D

REPLY [2 votes]: The statements in the book are not quite precise or correct.
In the first statement, by "non-Euclidean spaces" the author means differential manifolds that are not open subsets of $\mathbb{R}^n$, not  non-Euclidean geometry of curved Riemannian manifolds.  
The chain rule is that the differential of a composite function is the composition of the differentials.  This is not a group property as the author claims, since the differentials do not have to be invertible for the chain rule to hold. 
In the second statement, the inverse function theorem is not a group property.  The function may be from one manifold to another, so there is a groupoid not a group (for the invertible functions).  And the existence statement in the theorem is not a formal consquence of any group structure, it is more like solving a differential equation to go from a linear inverse of the function's differential, to a full compositional inverse of the function in a neighborhood of a nonsingular point.