Inverse function theorem

Summary

In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.

The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction mapping theorem.

Statements edit

For functions of a single variable, the theorem states that if   is a continuously differentiable function with nonzero derivative at the point  ; then   is injective (or bijective onto the image) in a neighborhood of  , the inverse is continuously differentiable near  , and the derivative of the inverse function at   is the reciprocal of the derivative of   at  :

 

It can happen that a function   may be injective near a point   while  . An example is  . In fact, for such a function, the inverse cannot be differentiable at  , since if   were differentiable at  , then, by the chain rule,  , which implies  . (The situation is different for holomorphic functions; see #Holomorphic inverse function theorem below.)

For functions of more than one variable, the theorem states that if f is a continuously differentiable function from an open subset   of   into  , and the derivative   is invertible at a point a (that is, the determinant of the Jacobian matrix of f at a is non-zero), then there exist neighborhoods   of   in   and   of   such that   and   is bijective.[1] Writing  , this means that the system of n equations   has a unique solution for   in terms of   when  . Note that the theorem does not say   is bijective onto the image where   is invertible but that it is locally bijective where   is invertible.

Moreover, the theorem says that the inverse function   is continuously differentiable, and its derivative at   is the inverse map of  ; i.e.,

 

In other words, if   are the Jacobian matrices representing  , this means:

 

The hard part of the theorem is the existence and differentiability of  . Assuming this, the inverse derivative formula follows from the chain rule applied to  . (Indeed,  ) Since taking the inverse is infinitely differentiable, the formula for the derivative of the inverse shows that if   is continuously   times differentiable, with invertible derivative at the point a, then the inverse is also continuously   times differentiable. Here   is a positive integer or  .

There are two variants of the inverse function theorem.[1] Given a continuously differentiable map  , the first is

  • The derivative   is surjective (i.e., the Jacobian matrix representing it has rank  ) if and only if there exists a continuously differentiable function   on a neighborhood   of   such   near  ,

and the second is

  • The derivative   is injective if and only if there exists a continuously differentiable function   on a neighborhood   of   such   near  .

In the first case (when   is surjective), the point   is called a regular value. Since  , the first case is equivalent to saying   is not in the image of critical points   (a critical point is a point   such that the kernel of   is nonzero). The statement in the first case is a special case of the submersion theorem.

These variants are restatements of the inverse functions theorem. Indeed, in the first case when   is surjective, we can find an (injective) linear map   such that  . Define   so that we have:

 

Thus, by the inverse function theorem,   has inverse near  ; i.e.,   near  . The second case (  is injective) is seen in the similar way.

Example edit

Consider the vector-valued function   defined by:

 

The Jacobian matrix is:

 

with Jacobian determinant:

 

The determinant   is nonzero everywhere. Thus the theorem guarantees that, for every point p in  , there exists a neighborhood about p over which F is invertible. This does not mean F is invertible over its entire domain: in this case F is not even injective since it is periodic:  .

Counter-example edit

 
The function   is bounded inside a quadratic envelope near the line  , so  . Nevertheless, it has local max/min points accumulating at  , so it is not one-to-one on any surrounding interval.

If one drops the assumption that the derivative is continuous, the function no longer need be invertible. For example   and   has discontinuous derivative   and  , which vanishes arbitrarily close to  . These critical points are local max/min points of  , so   is not one-to-one (and not invertible) on any interval containing  . Intuitively, the slope   does not propagate to nearby points, where the slopes are governed by a weak but rapid oscillation.

Methods of proof edit

As an important result, the inverse function theorem has been given numerous proofs. The proof most commonly seen in textbooks relies on the contraction mapping principle, also known as the Banach fixed-point theorem (which can also be used as the key step in the proof of existence and uniqueness of solutions to ordinary differential equations).[2][3]

Since the fixed point theorem applies in infinite-dimensional (Banach space) settings, this proof generalizes immediately to the infinite-dimensional version of the inverse function theorem[4] (see Generalizations below).

An alternate proof in finite dimensions hinges on the extreme value theorem for functions on a compact set.[5]

Yet another proof uses Newton's method, which has the advantage of providing an effective version of the theorem: bounds on the derivative of the function imply an estimate of the size of the neighborhood on which the function is invertible.[6]

A proof using successive approximation edit

To prove existence, it can be assumed after an affine transformation that   and  , so that  .

By the mean value theorem for vector-valued functions, for a function  ,  . Setting  , it follows that

 

Now choose   so that   for  . Suppose that   and define   inductively by   and  . The assumptions show that if   then

 .

In particular   implies  . In the inductive scheme   and  . Thus   is a Cauchy sequence tending to  . By construction   as required.

To check that   is C1, write   so that  . By the inequalities above,   so that  . On the other hand if  , then  . Using the geometric series for  , it follows that  . But then

 

tends to 0 as   and   tend to 0, proving that   is C1 with  .

The proof above is presented for a finite-dimensional space, but applies equally well for Banach spaces. If an invertible function   is Ck with  , then so too is its inverse. This follows by induction using the fact that the map   on operators is Ck for any   (in the finite-dimensional case this is an elementary fact because the inverse of a matrix is given as the adjugate matrix divided by its determinant). [1][7] The method of proof here can be found in the books of Henri Cartan, Jean Dieudonné, Serge Lang, Roger Godement and Lars Hörmander.

A proof using the contraction mapping principle edit

Here is a proof based on the contraction mapping theorem. Specifically, following T. Tao,[8] it uses the following consequence of the contraction mapping theorem.

Lemma — Let   denote an open ball of radius r in   with center 0. If   is a map such that   and there exists a constant   such that

 

for all   in  , then   is injective on   and  .

(More generally, the statement remains true if   is replaced by a Banach space.)

Basically, the lemma says that a small perturbation of the identity map by a contraction map is injective and preserves a ball in some sense. Assuming the lemma for a moment, we prove the theorem first. As in the above proof, it is enough to prove the special case when   and  . Let  . The mean value inequality applied to   says:

 

Since   and   is continuous, we can find an   such that

 

for all   in  . Then the early lemma says that   is injective on   and  . Then

 

is bijective and thus has an inverse. Next, we show the inverse   is continuously differentiable (this part of the argument is the same as that in the previous proof). This time, let   denote the inverse of   and  . For  , we write   or  . Now, by the early estimate, we have

 

and so  . Writing   for the operator norm,

 

As  , we have   and   is bounded. Hence,   is differentiable at   with the derivative  . Also,   is the same as the composition   where  ; so   is continuous.

It remains to show the lemma. First, the map   is injective on   since if  , then   and so

 ,

which is a contradiction unless  . (This part does not need the assumption  .) Next we show  . The idea is to note that this is equivalent to, given a point   in  , find a fixed point of the map

 

where   such that   and the bar means a closed ball. To find a fixed point, we use the contraction mapping theorem and checking that   is a well-defined strict-contraction mapping is straightforward. Finally, we have:   since

 

As might be clear, this proof is not substantially different from the previous one, as the proof of the contraction mapping theorem is by successive approximation.

Applications edit

Implicit function theorem edit

The inverse function theorem can be used to solve a system of equations

 

i.e., expressing   as functions of  , provided the Jacobian matrix is invertible. The implicit function theorem allows to solve a more general system of equations:

 

for   in terms of  . Though more general, the theorem is actually a consequence of the inverse function theorem. First, the precise statement of the implicit function theorem is as follows:[9]

  • given a map  , if  ,   is continuously differentiable in a neighborhood of   and the derivative of   at   is invertible, then there exists a differentiable map   for some neighborhoods   of   such that  . Moreover, if  , then  ; i.e.,   is a unique solution.

To see this, consider the map  . By the inverse function theorem,   has the inverse   for some neighborhoods  . We then have:

 

implying   and   Thus   has the required property.  

Giving a manifold structure edit

In differential geometry, the inverse function theorem is used to show that the pre-image of a regular value under a smooth map is a manifold.[10] Indeed, let   be such a smooth map from an open subset of   (since the result is local, there is no loss of generality with considering such a map). Fix a point   in   and then, by permuting the coordinates on  , assume the matrix   has rank  . Then the map   is such that   has rank  . Hence, by the inverse function theorem, we find the smooth inverse   of   defined in a neighborhood   of  . We then have

 

which implies

 

That is, after the change of coordinates by  ,   is a coordinate projection (this fact is known as the submersion theorem). Moreover, since   is bijective, the map

 

is bijective with the smooth inverse. That is to say,   gives a local parametrization of   around  . Hence,   is a manifold.   (Note the proof is quite similar to the proof of the implicit function theorem and, in fact, the implicit function theorem can be also used instead.)

More generally, the theorem shows that if a smooth map   is transversal to a submanifold  , then the pre-image   is a submanifold.[11]

Global version edit

The inverse function theorem is a local result; it applies to each point. A priori, the theorem thus only shows the function   is locally bijective (or locally diffeomorphic of some class). The next topological lemma can be used to upgrade local injectivity to injectivity that is global to some extent.

Lemma — [12][full citation needed][13] If   is a closed subset of a (second-countable) topological manifold   (or, more generally, a topological space admitting an exhaustion by compact subsets) and  ,   some topological space, is a local homeomorphism that is injective on  , then   is injective on some neighborhood of  .

Proof:[14] First assume   is compact. If the conclusion of the theorem is false, we can find two sequences   such that   and   each converge to some points   in  . Since   is injective on  ,  . Now, if   is large enough,   are in a neighborhood of   where   is injective; thus,  , a contradiction.

In general, consider the set  . It is disjoint from   for any subset   where   is injective. Let   be an increasing sequence of compact subsets with union   and with   contained in the interior of  . Then, by the first part of the proof, for each  , we can find a neighborhood   of   such that  . Then   has the required property.   (See also [15] for an alternative approach.)

The lemma implies the following (a sort of) global version of the inverse function theorem:

Inverse function theorem — [16] Let   be a map between open subsets of   or more generally of manifolds. Assume   is continuously differentiable (or is  ). If   is injective on a closed subset   and if the Jacobian matrix of   is invertible at each point of  , then   is injective in a neighborhood   of   and   is continuously differentiable (or is  ).

Note that if   is a point, then the above is the usual inverse function theorem.

Holomorphic inverse function theorem edit

There is a version of the inverse function theorem for holomorphic maps.

Theorem — [17][18] Let   be open subsets such that   and   a holomorphic map whose Jacobian matrix in variables   is invertible (the determinant is nonzero) at  . Then   is injective in some neighborhood   of   and the inverse   is holomorphic.

The theorem follows from the usual inverse function theorem. Indeed, let   denote the Jacobian matrix of   in variables   and   for that in  . Then we have  , which is nonzero by assumption. Hence, by the usual inverse function theorem,   is injective near   with continuously differentiable inverse. By chain rule, with  ,

 

where the left-hand side and the first term on the right vanish since   and   are holomorphic. Thus,   for each  .  

Similarly, there is the implicit function theorem for holomorphic functions.[19]

As already noted earlier, it can happen that an injective smooth function has the inverse that is not smooth (e.g.,   in a real variable). This is not the case for holomorphic functions because of:

Proposition — [19] If   is an injective holomorphic map between open subsets of  , then   is holomorphic.

Formulations for manifolds edit

The inverse function theorem can be rephrased in terms of differentiable maps between differentiable manifolds. In this context the theorem states that for a differentiable map   (of class  ), if the differential of  ,

 

is a linear isomorphism at a point   in   then there exists an open neighborhood   of   such that

 

is a diffeomorphism. Note that this implies that the connected components of M and N containing p and F(p) have the same dimension, as is already directly implied from the assumption that dFp is an isomorphism. If the derivative of F is an isomorphism at all points p in M then the map F is a local diffeomorphism.

Generalizations edit

Banach spaces edit

The inverse function theorem can also be generalized to differentiable maps between Banach spaces X and Y.[20] Let U be an open neighbourhood of the origin in X and   a continuously differentiable function, and assume that the Fréchet derivative   of F at 0 is a bounded linear isomorphism of X onto Y. Then there exists an open neighbourhood V of   in Y and a continuously differentiable map   such that   for all y in V. Moreover,   is the only sufficiently small solution x of the equation  .

There is also the inverse function theorem for Banach manifolds.[21]

Constant rank theorem edit

The inverse function theorem (and the implicit function theorem) can be seen as a special case of the constant rank theorem, which states that a smooth map with constant rank near a point can be put in a particular normal form near that point.[22] Specifically, if   has constant rank near a point  , then there are open neighborhoods U of p and V of   and there are diffeomorphisms   and   such that   and such that the derivative   is equal to  . That is, F "looks like" its derivative near p. The set of points   such that the rank is constant in a neighborhood of   is an open dense subset of M; this is a consequence of semicontinuity of the rank function. Thus the constant rank theorem applies to a generic point of the domain.

When the derivative of F is injective (resp. surjective) at a point p, it is also injective (resp. surjective) in a neighborhood of p, and hence the rank of F is constant on that neighborhood, and the constant rank theorem applies.

Polynomial functions edit

If it is true, the Jacobian conjecture would be a variant of the inverse function theorem for polynomials. It states that if a vector-valued polynomial function has a Jacobian determinant that is an invertible polynomial (that is a nonzero constant), then it has an inverse that is also a polynomial function. It is unknown whether this is true or false, even in the case of two variables. This is a major open problem in the theory of polynomials.

Selections edit

When   with  ,   is   times continuously differentiable, and the Jacobian   at a point   is of rank  , the inverse of   may not be unique. However, there exists a local selection function   such that   for all   in a neighborhood of  ,  ,   is   times continuously differentiable in this neighborhood, and   (  is the Moore–Penrose pseudoinverse of  ).[23]

See also edit

Notes edit

  1. ^ a b c Theorem 1.1.7. in Hörmander, Lars (2015). The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Classics in Mathematics (2nd ed.). Springer. ISBN 978-3-642-61497-2.
  2. ^ McOwen, Robert C. (1996). "Calculus of Maps between Banach Spaces". Partial Differential Equations: Methods and Applications. Upper Saddle River, NJ: Prentice Hall. pp. 218–224. ISBN 0-13-121880-8.
  3. ^ Tao, Terence (12 September 2011). "The inverse function theorem for everywhere differentiable maps". Retrieved 26 July 2019.
  4. ^ Jaffe, Ethan. "Inverse Function Theorem" (PDF).
  5. ^ Spivak 1965, pages 31–35
  6. ^ Hubbard, John H.; Hubbard, Barbara Burke (2001). Vector Analysis, Linear Algebra, and Differential Forms: A Unified Approach (Matrix ed.).
  7. ^ Cartan, Henri (1971). Calcul Differentiel (in French). Hermann. pp. 55–61. ISBN 978-0-395-12033-0.
  8. ^ Theorem 17.7.2 in Tao, Terence (2014). Analysis. II. Texts and Readings in Mathematics. Vol. 38 (Third edition of 2006 original ed.). New Delhi: Hindustan Book Agency. ISBN 978-93-80250-65-6. MR 3310023. Zbl 1300.26003.
  9. ^ Spivak 1965, Theorem 2-12.
  10. ^ Spivak 1965, Theorem 5-1. and Theorem 2-13.
  11. ^ "Transversality" (PDF). northwestern.edu.
  12. ^ One of Spivak's books (Editorial note: give the exact location).
  13. ^ Hirsch 1976, Ch. 2, § 1., Exercise 7. NB: This one is for a  -immersion.
  14. ^ Lemma 13.3.3. of Lectures on differential topology utoronto.ca
  15. ^ Dan Ramras (https://mathoverflow.net/users/4042/dan-ramras), On a proof of the existence of tubular neighborhoods., URL (version: 2017-04-13): https://mathoverflow.net/q/58124
  16. ^ Ch. I., § 3, Exercise 10. and § 8, Exercise 14. in V. Guillemin, A. Pollack. "Differential Topology". Prentice-Hall Inc., 1974. ISBN 0-13-212605-2.
  17. ^ Griffiths & Harris 1978, p. 18.
  18. ^ Fritzsche, K.; Grauert, H. (2002). From Holomorphic Functions to Complex Manifolds. Springer. pp. 33–36. ISBN 978-0-387-95395-3.
  19. ^ a b Griffiths & Harris 1978, p. 19.
  20. ^ Luenberger, David G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons. pp. 240–242. ISBN 0-471-55359-X.
  21. ^ Lang, Serge (1985). Differential Manifolds. New York: Springer. pp. 13–19. ISBN 0-387-96113-5.
  22. ^ Boothby, William M. (1986). An Introduction to Differentiable Manifolds and Riemannian Geometry (Second ed.). Orlando: Academic Press. pp. 46–50. ISBN 0-12-116052-1.
  23. ^ Dontchev, Asen L.; Rockafellar, R. Tyrrell (2014). Implicit Functions and Solution Mappings: A View from Variational Analysis (Second ed.). New York: Springer-Verlag. p. 54. ISBN 978-1-4939-1036-6.

References edit

  • Allendoerfer, Carl B. (1974). "Theorems about Differentiable Functions". Calculus of Several Variables and Differentiable Manifolds. New York: Macmillan. pp. 54–88. ISBN 0-02-301840-2.
  • Baxandall, Peter; Liebeck, Hans (1986). "The Inverse Function Theorem". Vector Calculus. New York: Oxford University Press. pp. 214–225. ISBN 0-19-859652-9.
  • Nijenhuis, Albert (1974). "Strong derivatives and inverse mappings". Amer. Math. Monthly. 81 (9): 969–980. doi:10.2307/2319298. hdl:10338.dmlcz/102482. JSTOR 2319298.
  • Griffiths, Phillip; Harris, Joseph (1978), Principles of Algebraic Geometry, John Wiley & Sons, ISBN 978-0-471-05059-9.
  • Hirsch, Morris W. (1976). Differential Topology. Springer-Verlag. ISBN 978-0-387-90148-0.
  • Protter, Murray H.; Morrey, Charles B. Jr. (1985). "Transformations and Jacobians". Intermediate Calculus (Second ed.). New York: Springer. pp. 412–420. ISBN 0-387-96058-9.
  • Renardy, Michael; Rogers, Robert C. (2004). An Introduction to Partial Differential Equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 337–338. ISBN 0-387-00444-0.
  • Rudin, Walter (1976). Principles of mathematical analysis. International Series in Pure and Applied Mathematics (Third ed.). New York: McGraw-Hill Book. pp. 221–223. ISBN 978-0-07-085613-4.
  • Spivak, Michael (1965). Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. San Francisco: Benjamin Cummings. ISBN 0-8053-9021-9.