# Inverse function theorem

In mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain. 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. In this case, the theorem gives 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.

## Statement of the theorem

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

${\bigl (}f^{-1}{\bigr )}'(b)={\frac {1}{f'(a)}}$ $J_{F^{-1}}(F(p))=[J_{F}(p)]^{-1}$ $J_{G\circ H}(p)=J_{G}(H(p))\cdot J_{H}(p).$ ## Example

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

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

## 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 $F:M\to N$ , if the derivative of $F$ ,

$dF_{p}:T_{p}M\to T_{F(p)}N$ $F|_{U}:U\to F(U)$ ### Banach manifolds

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

### 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. 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.