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 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 ƒ is a continuously differentiable function and ƒ has nonzero derivative at a, then ƒ is invertible in a neighborhood of a, the inverse is continuously differentiable, and

${\bigl (}f^{-1}{\bigr )}'(b)={\frac {1}{f'(a)}}$

where b = ƒ(a).

For functions of more than one variable, 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 at p.

Finally, the theorem says that

$J_{F^{-1}}(F(p))=[J_{F}(p)]^{-1}$

where $[\cdot ]^{-1}$ denotes matrix inverse and $J_{G}(q)$ is the Jacobian matrix of the function G at the point q.

This formula can also be derived from the chain rule. The chain rule states that for functions G and H which have total derivatives at H(p) and p respectively,

$J_{G\circ H}(p)=J_{G}(H(p))\cdot J_{H}(p).$

Letting G be F ^-1 and H be F, $G\circ H$ is the identity function, whose Jacobian matrix is also the identity. In this special case, the formula above can be solved for $J_{F^{-1}}(F(p))$. Note that the chain rule assumes the existence of total derivative of the inside function H, while the inverse function theorem proves that F^-1 has a total derivative at p.

The existence of an inverse function to F is equivalent to saying 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 and F(p), respectively.

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. Note that this is different than saying F is invertible over its entire image. In this example, F is not invertible because it is not injective (because $f(x,y)=f(x,y+2\pi )$.)

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 → 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.

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.

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.

See also

• Implicit function theorem

References

• Nijenhuis, Albert (1974). "Strong derivatives and inverse mappings". Amer. Math. Monthly. 81: 969–980. doi:10.2307/2319298.
• Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 337–338. ISBN 0-387-00444-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Rudin, Walter (1976). Principles of mathematical analysis. International Series in Pure and Applied Mathematics (Third ed.). New York: McGraw-Hill Book Co. pp. 221–223.