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.

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,

${\displaystyle J_{G\circ F}(p)=J_{G}(F(p))\cdot J_{F}(p)}$

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.

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.

Example

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

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

Then the Jacobian matrix is

${\displaystyle J_{F}(x,y)={\begin{bmatrix}{e^{x}\cos y}&{-e^{x}\sin y}\\{e^{x}\sin y}&{e^{x}\cos y}\\\end{bmatrix}}}$ 

and the determinant is

${\displaystyle \det J_{F}(x,y)=e^{2x}\cos ^{2}y+e^{2x}\sin ^{2}y=e^{2x}.\,\!}$ 

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.

References

• Albert Nijenhuis. "Strong derivatives and inverse mappings." American Mathematical Monthly. Vol. 81, 1974, Pages 969-980.
• Walter Rudin. Principles of Mathematical Analysis. Third Edition. McGraw-Hill, Inc., 1976, Pages 221-223.