Can the Inverse Function Theorem Be Applied to Complex Multivariable Functions?

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Discussion Overview

The discussion centers around the application of the inverse function theorem to complex multivariable functions, specifically functions mapping from R² to R. Participants explore the conditions under which the theorem can be applied and the implications of those conditions.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant questions the applicability of the inverse function theorem to functions like f(x,y) = -xye^(-(x²+y²)/2) and f(x,y) = 2x²+y²-xy-7y, expressing confusion about how to determine their invertibility.
  • Another participant points out that the inverse function theorem typically applies to functions from Rⁿ to Rⁿ, suggesting that the functions in question do not fit this criterion since they map from R² to R.
  • A participant expresses confusion about the task of showing whether the given functions are invertible using the inverse function theorem, asking for examples of applicable scenarios.
  • One reply indicates that the inverse function theorem may not apply directly, but mentions the rank theorem, which could provide a framework for understanding the situation if the Jacobian matrix's rank is considered.
  • This same participant elaborates that while the inverse function theorem does not apply directly to the original functions, it can be used in a modified context to derive auxiliary functions that are invertible, thus providing a structure theorem for the original functions.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the inverse function theorem to the specific functions discussed. There is no consensus on whether the theorem can be applied in this context, with some arguing it cannot while others suggest a modified application may be possible.

Contextual Notes

The discussion highlights limitations regarding the dimensionality of the functions involved and the conditions necessary for the inverse function theorem to hold. There is also mention of the need for the Jacobian matrix to be invertible, which is a critical assumption in applying the theorem.

DuskStar
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I've had a read through some of the topics about this but I am struggling to understand how to apply it.

(1) Is it possible to apply the inverse function theorem to a function like f(x,y)=-xye(-(x^2+y^2)/2) or f(x,y)=2x^2+y^2-xy-7y

(2) I am confused about how to compute the jacobian matrix when differentating only gives two terms and my matrix needs four?

Any help would be great
 
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What "inverse function theorem" are you referring to? The "inverse functions theorems" I know are from Rn to Rn. Here, your function is from R2 to R. An "inverse" would be from R to R2. I don't see any reasonable way to do that!
 
In that case I am completely confused. I was just given these examples and told to show whether they were invertible or not by the 'inverse function theorem'. Is it possible for someone to post any example where I could apply the inverse function theorem?
 
You originally said the question was " Is it possible to apply the inverse function theorem to a function like f(x,y)=-xye(-(x^2+y^2)/2) or f(x,y)=2x^2+y^2-xy-7y?"

Possibly the answer is "no"!
 
the inverse function theorem, as stated, applies only to functions between euclidean spaces of the same dimension, since the hypothesis, that the jacobian matrix be invertible, is otherwise false.

however it is a special case of the "rank theorem', which says, also in a special case, that if the rank of the jacobian matrix equals the dimension of the target, say n, (and the map is smooth), then in some smooth coordinate system, the map becomes projection on the last n coordinates.

so in this example of a map from R^2 to R, near any point where the partials are not both zero, the map can be expressed in some coordinate system as (x,y)-->y.

this theorem, also called the implicit function theorem, can be proved as a corollary of the inverse function theorem, by augmenting the given map as (f(x,y), x), or f(x,y),y), depending on which partial is non zero.

so in some sense the inverse function theorem can be applied to this situation. the result however gives that some auxiliary function is invertible, not the original f. for the original f it gives a structure theorem.
 

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