Below are theorems from Munkres' "Analysis on Manifolds". The proof of Theorem 7.4 on the right invokes the chain rule, stated on the left. The conditions of Theorem are somewhat strange and appear only to be needed so the chain rule's conditions are satisfied in the proof. However, even when I find a convoluted to make the appropriate substitutions to show why the chain rule can be invoked, I am left with many conditions stated that are never actually used, or are redundant. Take a look:
What's going on? The theorem seems like it's badly botched, especially considering that the conditions can actually just be reduced to:
- f and g are functions between euclidean spaces.
- f and g are differentiable at a and f(a), respectively
- g is f's left inverse on a neighborhood of a in f's domain.
- ∴ Dg(f(a)) = [Df(a)]⁻¹
Is this a correct way to produce a simplified statement of the theorem which is equivalent to Munkres'?