Why Is the Inverse Function Theorem by Spivak Difficult to Follow?

[tjkubo]

I'm having trouble following the proof of the IFT by Spivak. The statement of the theorem was posted in a similar thread:
https://www.physicsforums.com/showthread.php?t=319924

He says, "If the theorem is true for $$\lambda^{-1} \circ f$$, it is clearly true for $$f$$. Therefore we may assume at the outset that $$\lambda$$ is the identity."

These statements are not clear to me, so if anyone can provide a little more explanation, that would be helpful.

 

[jambaugh]

I'm not sure I follow the chain of reasoning from the thread. But the basic proof is straightforward.

Apply the chain rule to the derivative (w.r.t. y) of $$[f\circ f^{-1}](\mathbf{y})=\mathbf{y}$$
you get:
$$[Df]\circ f(\mathbf{y})\cdot Df^{-1}(\mathbf{y}) = \mathbf{1}$$
thence
$$Df^{-1}(\mathbf{y}) = [Df(f^{-1}(\mathbf{y}))]^{-1}$$

This works for single valued functions and for functions of many variables (treated as a vector valued function of a vector.)