Inverse function theorem

From Wikipedia, the free encyclopedia

In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. The theorem can be generalized to maps defined on manifolds, and on infinite dimensional Banach spaces (and Banach manifolds). Loosely, a C1 function F is invertible at a point p if its Jacobian JF(p) is invertible.

Contents

[edit] Statement of the theorem

More precisely, the theorem states that if the total derivative of a continuously differentiable function F defined from an open set U of Rn into Rn is invertible at a point p (i.e., the Jacobian determinant of F at p is nonzero), then F is an invertible function near p. That is, an inverse function to F exists in some neighborhood of F(p). Moreover, the inverse function F -1 is also continuously differentiable. In the infinite dimensional case it is required that the Frechet derivative have a bounded inverse near p.

The Jacobian matrix of F -1 at F(p) is then the inverse of the Jacobian of F, evaluated at p. This can be understood as a special case of the chain rule, which states that for linear transformations f and g,

J_{G \circ F} (p) = J_G (F(p)) \cdot J_F (p)

where J denotes the corresponding Jacobian matrix.

The conclusion of the theorem is that the system of n equations yi = Fj(x1,...,xn) can be solved for x1,...,xn in terms of y1,...,yn if we restrict x and y to small enough neighborhoods of p.

[edit] Example

Consider the vector-valued function F from R2 to R2 defined by

\mathbf{F}(x,y)= \begin{bmatrix} {e^x \cos y}\\ {e^x \sin y}\\ \end{bmatrix}.

Then the Jacobian matrix is

J_F(x,y)= \begin{bmatrix} {e^x \cos y} & {-e^x \sin y}\\ {e^x \sin y} & {e^x \cos y}\\ \end{bmatrix}

and the determinant is

\det J_F(x,y)= e^{2x} \cos^2 y + e^{2x} \sin^2 y= e^{2x}. \,\!

The determinant e2x is nonzero everywhere. By the theorem, for every point p in R2, there exists a neighborhood about p over which F is invertible.

[edit] Notes on methods of proof

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. (This theorem can also be used as the key step in the proof of existence and uniqueness of solutions to ordinary differential equations.) Since this theorem applies in infinite-dimensional (Banach space) settings, it is the tool used in proving the infinite-dimensional version of the inverse function theorem (see "Generalizations", below).

An alternate proof (which works only in finite dimensions) instead uses as the key tool the extreme value theorem for functions on a compact set.[1]

Yet another proof uses Newton's Method, which has the advantage of providing an effective version of the theorem. That is, given specific bounds on the derivative of the function, an estimate of the size of the neighborhood on which the function is invertible can be obtained.[2]

[edit] Generalizations

[edit] Manifolds

The inverse function theorem can be generalized to differentiable maps between differentiable manifolds. In this context the theorem states that for a differentiable map F : M → N, if the derivative of F,

(dF)p : TpM → TF(p)N

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

F|U : U → F(U)

is a diffeomorphism. Note that this implies that M and N must have the same dimension.

If the derivative of F is an isomorphism at all points p in M then the map F is a local diffeomorphism.

[edit] Banach spaces

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

In the simple case where the function is a bijection between X and Y, the function has a continuous inverse. This follows immediately from the open mapping theorem.

[edit] Banach manifolds

These two directions of generalization can be combined in the inverse function theorem for Banach manifolds.[3]

[edit] Constant rank theorem

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 locally constant rank near a point can be put in a particular normal form near that point.[4] When the derivative of F is invertible at a point p, it is also invertible in a neighborhood of p, and hence the rank of the derivative is constant, so the constant rank theorem applies.

[edit] See also

[edit] Notes

  1. ^ Michael Spivak, Calculus on Manifolds.
  2. ^ John H. Hubbard and Barbara Burke Hubbard, Vector Analysis, Linear Algebra, and Differential Forms: a unified approach, Matrix Editions, 2001.
  3. ^ Serge Lang, Differential and Riemannian Manifolds, Springer, 1995, ISBN 0387943382.
  4. ^ Wiilliam M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, 2002, ISBN 0121160513.

[edit] References

  • Nijenhuis, Albert (1974). "Strong derivatives and inverse mappings". Amer. Math. Monthly 81: 969–980. 
  • Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations, Second edition, Texts in Applied Mathematics 13, New York: Springer-Verlag, 337–338. ISBN 0-387-00444-0. 
  • Rudin, Walter (1976). Principles of mathematical analysis, Third edition, International Series in Pure and Applied Mathematics, New York: McGraw-Hill Book Co., 221–223.