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.

StatementEdit

For functions of a single variable, the theorem states that if   is a continuously differentiable function with nonzero derivative at the point a; then   is invertible in a neighborhood of a, the inverse is continuously differentiable, and the derivative of the inverse function at   is the reciprocal of the derivative of   at  :

 

An alternate version, which assumes that   is continuous and injective near a, and differentiable at a with a non-zero derivative, will also result in   being invertible near a, with an inverse that's similarly continuous and injective , and where the above formula would apply as well.

As a corollary, we see clearly that if   is  -th differentiable, with nonzero derivative at the point a, then   is invertible in a neighborhood of a, the inverse is also  -th differentiable. Here   is a positive integer or  .

For functions of more than one variable, the theorem states that if F is a continuously differentiable function from an open set of   into  , and the total derivative is invertible at a point p (that is, the Jacobian determinant of F at p is non-zero), then F is invertible near p: an inverse function to F is defined on some neighborhood of  . Writing  , this means that the system of n equations   has a unique solution for   in terms of  , provided that we restrict x and y to small enough neighborhoods of p and q, respectively.

Finally, the theorem says that the inverse function   is continuously differentiable, and its Jacobian derivative at   is the matrix inverse of the Jacobian of F at p:

 
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  :
 

ExampleEdit

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-exampleEdit

 
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 proofEdit

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).[1][2]

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[3] (see Generalizations below).

An alternate proof in finite dimensions hinges on the extreme value theorem for functions on a compact set.[4]

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.[5]

A proof using successive approximationEdit

The inverse function theorem states that if   is a C1 vector-valued function on an open set  , then   if and only if there is a C1 vector-valued function   defined near   with   near   and   near  . This 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. Taking derivatives, it follows that  .

The chain rule implies that the matrices   and   are each inverses. Continuity of   and   means that they are homeomorphisms that are each inverses locally. To prove existence, it can be assumed after an affine transformation that   and  , so that  .

By the fundamental theorem of calculus if   is a C1 function,  , so that  . 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). [6][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 principleEdit

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 the 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  . 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 it 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.

ApplicationsEdit

Implicit function theoremEdit

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

 

i.e., expressing   as functions of  , provided the Jacobian matrices are 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  .

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 structureEdit

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. More generally, the theorem shows that, given a smooth map  , if   is transversal to   a submanifold, then the pre-image   is a submanifold.[10]

GeneralizationsEdit

ManifoldsEdit

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.

Banach spacesEdit

The inverse function theorem can also be generalized to differentiable maps between Banach spaces X and Y.[11] 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  .

Banach manifoldsEdit

These two directions of generalization can be combined in the inverse function theorem for Banach manifolds.[12]

Constant rank theoremEdit

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.[13] 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 neighbourhood 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.

Holomorphic functionsEdit

If a holomorphic function F is defined from an open set U of   into  , and the Jacobian matrix of complex derivatives is invertible at a point p, then F is an invertible function near p. This follows immediately from the real multivariable version of the theorem. One can also show that the inverse function is again holomorphic.[14]

Polynomial functionsEdit

If it would be 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.

SelectionsEdit

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  ).[15]

See alsoEdit

NotesEdit

  1. ^ 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.
  2. ^ Tao, Terence (September 12, 2011). "The inverse function theorem for everywhere differentiable maps". Retrieved 2019-07-26.
  3. ^ Jaffe, Ethan. "Inverse Function Theorem" (PDF).
  4. ^ Spivak, Michael (1965). Calculus on Manifolds. Boston: Addison-Wesley. pp. 31–35. ISBN 0-8053-9021-9.
  5. ^ Hubbard, John H.; Hubbard, Barbara Burke (2001). Vector Analysis, Linear Algebra, and Differential Forms: A Unified Approach (Matrix ed.).
  6. ^ Hörmander, Lars (2015). The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Classics in Mathematics (2nd ed.). Springer. p. 10. ISBN 9783642614972.
  7. ^ Cartan, Henri (1971). Calcul Differentiel (in French). Hermann. pp. 55–61. ISBN 9780395120330.
  8. ^ Theorem 17.7. 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, Theorem 2-12.
  10. ^ https://sites.math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf
  11. ^ Luenberger, David G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons. pp. 240–242. ISBN 0-471-55359-X.
  12. ^ Lang, Serge (1985). Differential Manifolds. New York: Springer. pp. 13–19. ISBN 0-387-96113-5.
  13. ^ 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.
  14. ^ Fritzsche, K.; Grauert, H. (2002). From Holomorphic Functions to Complex Manifolds. Springer. pp. 33–36.
  15. ^ 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.

ReferencesEdit