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)'(f(a)) = \frac{1}{f'(a)},

where notationally the left side refers to the derivative of the inverse function evaluated at f(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 of $\mathbb{R}^n$ into $\mathbb{R}^n$ 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 $F^{-1}$ 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

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

where $[\cdot]^{-1}$ denotes matrix inverse and $J_F(p)$ is the Jacobian matrix of the function $F$ at the point $p$ . 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 $y_i = F_i(x_1, \dots, x_n)$ can be solved for $x_1, \dots, x_n$ in terms of $y_1, \dots, y_n$ if we restrict $x$ and $y$ to small enough neighborhoods of $p$ and $F(p)$ , respectively.

Example

Consider the vector-valued function $F$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ 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 $\mathbb{R}^2$ , 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.^[1] 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.^[2]

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 differential of $F$ ,

dF_p: T_p M \to 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 \to 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.

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 $F: U \to Y$ be continuously differentiable and assume that the derivative $dF_0: X \to Y$ 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 $G: V \to X$ such that $F(G(y)) = y$ for all $y$ in $V$ . Moreover, $G(y)$ is the only sufficiently small solution $x$ of the equation $F(x) = y$ .

Banach manifolds

These two directions of generalization can be combined in the inverse function theorem for Banach manifolds.^[3]

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.^[4] Specifically, if $F:M\to N$ has constant rank near a point $p\in M$ , then there are open neighborhoods $U$ of $p$ and $V$ of $F(p)$ and there are diffeomorphisms $u:T_pM\to U$ and $v:T_{F(p)}N\to V$ such that $F(U)\subseteq V$ and such that the derivative $dF_p:T_pM\to T_{F(p)}N$ is equal to $v^{-1}\circ F\circ u$ . That is, $F$ "looks like" its derivative near $p$ . 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 $F$ is injective (resp. surjective) at a point $p$ , it is also injective (resp. surjective) in a neighborhood of $p$ , and hence the rank of $F$ is constant on that neighborhood, so the constant rank theorem applies.

Holomorphic Functions

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 $\mathbb{C}^n$ into $\mathbb{C}^n$ , 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.^[5]

Notes

↑ Michael Spivak, Calculus on Manifolds.
↑ John H. Hubbard and Barbara Burke Hubbard, Vector Analysis, Linear Algebra, and Differential Forms: a unified approach, Matrix Editions, 2001.
↑ Lang 1995, Lang 1999, pp. 15–19, 25–29.
↑ William M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised Second Edition, Academic Press, 2002, ISBN 0-12-116051-3.
↑ K. Fritzsche, H. Grauert, "From Holomorphic Functions to Complex Manifolds", Springer-Verlag, (2002). Page 33.

References

Lang, Serge (1995). Differential and Riemannian Manifolds. Springer. ISBN 0-387-94338-2.
Lang, Serge (1999). Fundamentals of Differential Geometry. Graduate Texts in Mathematics. New York: Springer. ISBN 978-0-387-98593-0.
Nijenhuis, Albert (1974). "Strong derivatives and inverse mappings". Amer. Math. Monthly 81 (9): 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.
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.

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