Spivak Inverse Function Theorem Proof

The proof depends on (a) because it allows us to simplify the problem by considering only functions with derivative at a equal to the identity.
  • #1
krcmd1
62
0
On p. 36 of "Calculus on Manifolds" Spivak writes:

"If the theorem is true for ([tex]\lambda[/tex][tex]^{-1}[/tex])[tex]\circ[/tex]f , it is clearly true for f."

This far I understand. However, he next says:

"Therefore we may assume at the outset that [tex]\lambda[/tex] is the identity."

I don't understand how this follows, since he previously defined [tex]\lambda[/tex] = Df(x).

I would appreciate someone adding a bit more explanation here.

Thank you.
 
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  • #2
What he is saying is that since, whenever some theorem is true for [itex]\lambda^{-1}\circle f[/itex], it is true for f, we can work with f rather than [itex]\lambda^{-1}\circle f[/itex]- that is that we can take [itex]\lambda= 1[/itex]. What you need to do is look at why "if the theorem is true for [itex]\lambda^{-1}\circle f[/itex], it is clearly true for f.[/itex]
 
  • #3
I can understand why

a)if this particular theorem is true for [tex]\lambda^{-1}\circ[/tex]f it is true for f, but

b) is it true as your posting suggests that any theorem true for [tex]\lambda^{-1}\circ[/tex]f is true for f?

and

c) how does his proof depend upon (a)? I mean, how does the subsequent argument depend upon (a)? I understand that [tex]\lambda[/tex] is a linear transformation with non-zero determinant. Doesn't that already imply that f(x+a) -f(a) <> 0 in some neighborhood of a?

I hate to look a gift horse in the mouth but I'm studying this stuff on my own with no one to talk to - not in a class.

Thank you all.
 
  • #4
I know this thread is old, but I wanted to put in my two cents since I had trouble with this at first as well.
Suppose the theorem is true for any function with derivative at a equal to the identity function and suppose we have a function f with λ = Df(a) not necessarily the identity. As the author points out, [itex]\lambda^{-1} \circ f[/itex] is a continuously differentiable function with derivative at a equal to the identity, so the theorem is true for [itex]\lambda^{-1} \circ f[/itex] by the assumption above. Hence, the theorem is true for f.
 

Related to Spivak Inverse Function Theorem Proof

What is the Spivak Inverse Function Theorem Proof?

The Spivak Inverse Function Theorem Proof is a mathematical proof that states conditions for a function to have an inverse. It is named after mathematician Michael Spivak and is often used in calculus and differential geometry.

What are the conditions for a function to have an inverse according to the Spivak Inverse Function Theorem Proof?

The conditions are that the function must be continuous, differentiable, and have a non-zero derivative at a given point. These conditions ensure that the inverse function exists and is also continuous and differentiable.

How does the Spivak Inverse Function Theorem Proof differ from other inverse function theorems?

The Spivak Inverse Function Theorem Proof is more general and rigorous compared to other inverse function theorems, such as the Implicit Function Theorem. It also provides conditions for the existence of an inverse function at a specific point, rather than just a local inverse around that point.

What is the significance of the Spivak Inverse Function Theorem Proof in mathematics?

The Spivak Inverse Function Theorem Proof is significant because it provides a rigorous and general method for determining the existence of an inverse function. It is also used in many applications, such as optimization problems and differential equations.

What are some common applications of the Spivak Inverse Function Theorem Proof?

The Spivak Inverse Function Theorem Proof is commonly used in areas such as optimization, differential equations, and physics. It is also used in the study of manifolds and smooth functions in differential geometry.

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