Inverse function theorem

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. Loosely, a C¹ function F is invertible at a point p if its Jacobian J_F(p) is invertible.

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 y_i = F_j(x₁,...,x_n) can be solved for x₁,...,x_n in terms of y₁,...,y_n if we restrict x and y to small enough neighborhoods of p.

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 e^2x is nonzero everywhere. By the theorem, for every point p in R², there exists a neighborhood about p over which F is invertible.

Generalizations

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 : T_pM → T_F(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.

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. The first Banach space version of the inverse function theorem was proved by Lawrence Graves in 1950.^{[citation needed]}

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.

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. Texts in Applied Mathematics 13 (Second edition ed.). New York: Springer-Verlag. pp. 337–338. ISBN 0-387-00444-0. {{cite book}}: |edition= has extra text (help)CS1 maint: multiple names: authors list (link)
Rudin, Walter (1976). Principles of mathematical analysis. International Series in Pure and Applied Mathematics (Third edition ed.). New York: McGraw-Hill Book Co. pp. 221–223. {{cite book}}: |edition= has extra text (help)