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.)