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 is a continuously differentiable function with nonzero derivative at the point , then is invertible in a neighborhood of , the inverse is continuously differentiable, and
where notationally the left side refers to the derivative of the inverse function evaluated at its value f(a). For functions of more than one variable, the theorem states that if the total derivative of a continuously differentiable function defined from an open set of into is invertible at a point (i.e., the Jacobian determinant of at is non-zero), then is an invertible function near . That is, an inverse function to exists in some neighborhood of . Moreover, the inverse function is also continuously differentiable. In the infinite dimensional case it is required that the Fréchet derivative have a bounded inverse at . Finally, the theorem says that
where denotes matrix inverse and is the Jacobian matrix of the function at the point . This formula can also be derived from the chain rule. The chain rule states that for functions and which have total derivatives at and respectively,
Letting be and be , is the identity function, whose Jacobian matrix is also the identity. In this special case, the formula above can be solved for . Note that the chain rule assumes the existence of total derivative of the inside function , while the inverse function theorem proves that has a total derivative at . The existence of an inverse function to is equivalent to saying that the system of equations can be solved for in terms of if we restrict and to small enough neighborhoods of and , respectively.
Consider the vector-valued function from to defined by
Then the Jacobian matrix is
and the determinant is
The determinant is nonzero everywhere. By the theorem, for every point in , there exists a neighborhood about over which is invertible. Note that this is different than saying is invertible over its entire image. In this example, is not invertible because it is not injective (because ).
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.
is a linear isomorphism at a point in then there exists an open neighborhood of such that
The inverse function theorem can also be generalized to differentiable maps between Banach spaces. Let and be Banach spaces and an open neighbourhood of the origin in . Let be continuously differentiable and assume that the derivative of at 0 is a bounded linear isomorphism of onto . Then there exists an open neighbourhood of in and a continuously differentiable map such that for all in . Moreover, is the only sufficiently small solution of the equation .
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 constant rank near a point can be put in a particular normal form near that point. Specifically, if has constant rank near a point , then there are open neighborhoods of and of and there are diffeomorphisms and such that and such that the derivative is equal to . That is, "looks like" its derivative near . Semicontinuity of the rank function implies that the set of points near which the derivative has constant rank is an open dense subset of the domain of the map. So the constant rank theorem applies "generically" across the domain.
When the derivative of is injective (resp. surjective) at a point , it is also injective (resp. surjective) in a neighborhood of , and hence the rank of is constant on that neighborhood, so the constant rank theorem applies.
If the Jacobian (in this context the matrix formed by the complex derivatives) of a holomorphic function , defined from an open set of into , is invertible at a point , then is an invertible function near . This follows immediately from the theorem above. One can also show, that this inverse is again a holomorphic function.
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