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 with nonzero derivative at the point a, then ƒ is invertible in a neighborhood of a, the inverse is continuously differentiable, and
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 non-zero), 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 is also continuously differentiable. In the infinite dimensional case it is required that the Fréchet derivative have a bounded inverse at p.
Finally, the theorem says that
where denotes matrix inverse and 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,
Letting G be F -1 and H be F, 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 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 yi = Fj(x1,...,xn) can be solved for x1,...,xn in terms of y1,...,yn if we restrict x and y to small enough neighborhoods of p and F(p), respectively.
Consider the vector-valued function F from R2 to R2 defined by
Then the Jacobian matrix is
and the determinant is
The determinant e2x 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 .)
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).
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.
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 : TpM → TF(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 at p.
If the derivative of F is an isomorphism at all points p in M then the map F is a local diffeomorphism.
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.
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.
If the Jacobian (in this context the matrix formed by the complex derivatives) of a holomorphic function F, defined from an open set U of Cn into Cn , is invertible at a point p, then F is an invertible function near p. This follows immediately from the theorem above. One can also show, that this inverse is again a holomorphic function.
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