Open main menu

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.

Contents

Statement of the theoremEdit

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

 

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

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

 

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   in  , there exists a neighborhood about   over which   is invertible. This does not mean   is invertible over its entire domain: in this case   is not even injective since it is periodic:  .

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]

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 (see "Generalizations", below).

An alternate proof in finite dimensions hinges on the extreme value theorem for functions on a compact set.[2]

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.[3]

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   and   must have the same dimension at  . If the derivative of   is an isomorphism at all points   in   then the map   is a local diffeomorphism.

Banach spacesEdit

The inverse function theorem can also be generalized to differentiable maps between Banach spaces   and  .[4] Let   be an open neighbourhood of the origin in   and   a continuously differentiable function, and assume that the Fréchet derivative   of   at 0 is a bounded linear isomorphism of   onto  . Then there exists an open neighbourhood   of   in   and a continuously differentiable map   such that   for all   in  . Moreover,   is the only sufficiently small solution   of the equation  .

Banach manifoldsEdit

These two directions of generalization can be combined in the inverse function theorem for Banach manifolds.[5]

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.[6] Specifically, if   has constant rank near a point  , then there are open neighborhoods   of   and   of   and there are diffeomorphisms   and   such that   and such that the derivative   is equal to  . That is,   "looks like" its derivative near  . Semicontinuity of the rank function implies that there is an open dense subset of the domain of   on which the derivative has constant rank. Thus the constant rank theorem applies to a generic point of the domain.

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

Holomorphic functionsEdit

If a holomorphic function   is defined from an open set   of   into  , and the Jacobian matrix of complex derivatives is invertible at a point  , then   is an invertible function near  . This follows immediately from the real multivariable version of the theorem. One can also show that the inverse function is again holomorphic.[7]

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. ^ Spivak, Michael (1965). Calculus on Manifolds. Boston: Addison-Wesley. pp. 31–35. ISBN 0-8053-9021-9.
  3. ^ Hubbard, John H.; Hubbard, Barbara Burke (2001). Vector Analysis, Linear Algebra, and Differential Forms: A Unified Approach (Matrix ed.).
  4. ^ Luenberger, David G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons. pp. 240–242. ISBN 0-471-55359-X.
  5. ^ Lang, Serge (1985). Differential Manifolds. New York: Springer. pp. 13–19. ISBN 0-387-96113-5.
  6. ^ 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.
  7. ^ Fritzsche, K.; Grauert, H. (2002). From Holomorphic Functions to Complex Manifolds. Springer. pp. 33–36.

ReferencesEdit