Why Can't the Inverse Function Theorem Be Applied to a Function from R^n to R^m?

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Discussion Overview

The discussion centers on the limitations of applying the inverse function theorem to functions mapping from Rn to Rm, particularly focusing on the implications of the Jacobian not being a square matrix and the topological constraints involved.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants note that the Jacobian being non-square is a fundamental reason the inverse function theorem cannot be applied.
  • Others argue that applying the theorem would imply a homeomorphism between open sets of Rn and Rm, which is not possible due to topological reasons.
  • A participant provides an example where n=1 and m=2 to illustrate the impossibility of such a homeomorphism.
  • There is a question raised about the bijectivity of the function in the context of the inverse function theorem, with a participant asserting that having an inverse implies bijectivity.
  • Another participant suggests demonstrating that a map cannot be one-to-one from a higher-dimensional space to a lower-dimensional space as an alternative approach to understanding the limitations.

Areas of Agreement / Disagreement

Participants generally agree on the impossibility of applying the inverse function theorem in this context due to the non-square Jacobian and topological constraints, but there are differing views on the implications and alternative approaches to understanding this issue.

Contextual Notes

Some assumptions about the nature of the functions and their mappings are not explicitly stated, and the discussion does not resolve the nuances of bijectivity in relation to the inverse function theorem.

yifli
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I know it is impossible to apply inverse function theorem on a function from R^n to R^m because the Jacobian is not a square matrix.
Is there any other reason why this is impossible?
 
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Hi yifli! :smile:

If the inverse function theorem were applicable then we would obtain a homeomorphism between an open set of [itex]\mathbb{R}^n[/itex] and an open set of [itex]\mathbb{R}^m[/tex]. But it's a result from topology that such an open set can not be homeomorphic.<br /> <br /> For example if n=1 and m=2, then there would exist a homeomorphism between an interval ]a,b[ and an open disk, but such an homeomorphism would never exist.<br /> <br /> Does this answer your question?[/itex]
 
micromass said:
Hi yifli! :smile:

If the inverse function theorem were applicable then we would obtain a homeomorphism between an open set of [itex]\mathbb{R}^n[/itex] and an open set of [itex]\mathbb{R}^m[/tex]. But it's a result from topology that such an open set can not be homeomorphic.<br /> <br /> For example if n=1 and m=2, then there would exist a homeomorphism between an interval ]a,b[ and an open disk, but such an homeomorphism would never exist.<br /> <br /> Does this answer your question?[/itex]
[itex] <br /> Yes. thank you so much![/itex]
 
micromass said:
Hi yifli! :smile:

If the inverse function theorem were applicable then we would obtain a homeomorphism between an open set of [itex]\mathbb{R}^n[/itex] and an open set of [itex]\mathbb{R}^m[/itex]. But it's a result from topology that such an open set can not be homeomorphic.

Actually I have another question regarding your answer:
A homeomorphism f is a bijective mapping which is continuous and has a continuous inverse. Inverse function theorem guarantees that f is continuous and so is its inverse, but what about bijectivity?
 
yifli said:
Actually I have another question regarding your answer:
A homeomorphism f is a bijective mapping which is continuous and has a continuous inverse. Inverse function theorem guarantees that f is continuous and so is its inverse, but what about bijectivity?

If it has an inverse, it is automatically bijective. In fact, this is an equivalence.
 
yifli said:
I know it is impossible to apply inverse function theorem on a function from R^n to R^m because the Jacobian is not a square matrix.
Is there any other reason why this is impossible?

If you don't want to use the chain rule then try to show that the map can not be 1 to 1 from the higher dimensional space to the lower dimensional space.
 

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