Abstract Nonsense

Crushing one theorem at a time

The Inverse Function Theorem (Preliminaries)

Point of Post: In this post we give motivation for the inverse function theorem.

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In this post we discuss one of the most fundamental analytic-geometric facts in all of multivariable analysis–the inverse function theorem. The theorem really has its humble roots back in single variable analysis with an observation about regular points (points where the derivative is non-zero) of continuously differentiable functions. Namely, it is a common theorem that if f:(a,b)\to\mathbb{R} is continuously differentiable and f'(c)\ne 0 for some c\in(a,b)  then there exists some neighborhood U\subseteq (a,b) containing c for which f is bijective and it’s inverse is continuously differentiable and moreover that \displaystyle \left(f^{-1}\right)'(f(c))=\frac{1}{f'(c)}. There, the theorem was easy to prove (see any basic analysis textbook for a proof). So, since we are doing multivariable analysis an obvious question is “does this result extend to maps f:\mathbb{R}^n\to\mathbb{R}^m?” Well, the first problem in answering this question is formulating exactly what this ‘theorem’ would say in higher dimensions. Let’s  rephrase this theorem in a language a little more amenable to total derivatives. We begin with that f'(c)\ne 0 means. In particular,  (using the notation used above) we see that since D_f(c)(x)=f'(c)x we have that f'(c)\ne 0 if and only if D_f(c)\in\text{GL}\left(\mathbb{R}\right). Thus, it seems that the natural extension would be we want to consider f:U\to\mathbb{R}^m, with U\subseteq\mathbb{R}^n open, with some distinguished point c\in U such that D_f(c):\mathbb{R}^n\to\mathbb{R}^m is an isomorphism. In particular we should make the concession that we would like to only consider maps (with the above notation) where m=n. From this, we can see that we have a visually similar condition that takes f'(c)\ne 0 to (recalling that we are only considering maps \mathbb{R}^n\to\mathbb{R}^n) the condition \det\text{Jac}_f(c)\ne 0. It’s pretty intuitive that we should replace continuously differentiable with C^1(U) (in the multivariable sense). Lastly, we see that \displaystyle \left(f^{-1}\right)'(f(c))=\frac{1}{f'(c)} seems naturally translatable to D_{f^{-1}}(x)=D_f^{-1}(f^{-1}(x)) or, in the more common form, \text{Jac}_{f^{-1}}(x)=\text{Jac}_f^{-1}(f^{-1}(x)). Thus, we can finally create a single-variable to multivariable dictionary for this theorem

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\begin{array}{c|c}\mathbb{R}\to\mathbb{R} & \mathbb{R}^n\to\mathbb{R}^n\\ \hline & \text{ }\\ \text{continuously differentiable} & C^1(U)\\ & \\ f'(c)\ne 0 & D_f(c)\in\text{GL}\left(\mathbb{R}^n\right)\\ & \\ \displaystyle \left(f^{-1}\right)'(f(c))=\frac{1}{f'(c)} & D_{f^{-1}}(f(c))=D_f^{-1}(c)\end{array}

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1.  Spivak, Michael. Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus. New York: W.A. Benjamin, 1965. Print.

2. Apostol, Tom M. Mathematical Analysis. Reading, MA: Addison-Wesley Pub., 1974. Print.


September 8, 2011 - Posted by | Analysis | , , , ,


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