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In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. 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, giving 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 fdisplaystyle f is a continuously differentiable function with nonzero derivative at the point adisplaystyle a, then fdisplaystyle f is invertible in a neighborhood of adisplaystyle a, the inverse is continuously differentiable, and the derivative of the inverse function at b=f(a)displaystyle b=f(a) is the reciprocal of the derivative of fdisplaystyle f at adisplaystyle a:

(f−1)′(b)=1f′(a).displaystyle bigl (f^-1bigr )'(b)=frac 1f'(a).

For functions of more than one variable, the theorem states that if Fdisplaystyle F is a continuously differentiable function from an open set of Rndisplaystyle mathbb R ^n into Rndisplaystyle mathbb R ^n, and the total derivative is invertible at a point pdisplaystyle p (i.e., the Jacobian determinant of Fdisplaystyle F at pdisplaystyle p is non-zero), then Fdisplaystyle F is invertible near pdisplaystyle p: an inverse function to Fdisplaystyle F is defined on some neighborhood of q=F(p)displaystyle q=F(p).
Writing F=(F1,…,Fn)displaystyle F=(F_1,ldots ,F_n), this means the system of n equations yi=Fi(x1,…,xn)displaystyle y_i=F_i(x_1,dots ,x_n) has a unique solution for x1,…,xndisplaystyle x_1,dots ,x_n in terms of y1,…,yndisplaystyle y_1,dots ,y_n, provided we restrict xdisplaystyle x and ydisplaystyle y to small enough neighborhoods of pdisplaystyle p and qdisplaystyle q, respectively.
In the infinite dimensional case, the theorem requires the extra hypothesis that the Fréchet derivative of Fdisplaystyle F at pdisplaystyle p has a bounded inverse.

Finally, the theorem says that the inverse function F−1displaystyle F^-1 is continuously differentiable, and its Jacobian derivative at q=F(p)displaystyle q=F(p) is the matrix inverse of the Jacobian of Fdisplaystyle F at pdisplaystyle p:

JF−1(q)=[JF(p)]−1.displaystyle J_F^-1(q)=[J_F(p)]^-1.

The hard part of the theorem is the existence and differentiability of F−1displaystyle F^-1. Assuming this, the inverse derivative formula follows from the chain rule applied to F−1∘F=iddisplaystyle F^-1circ F=textid:

I=JF−1∘F(p) = JF−1(F(p))⋅JF(p) = JF−1(q)⋅JF(p).displaystyle I=J_F^-1circ F(p) = J_F^-1(F(p))cdot J_F(p) = J_F^-1(q)cdot J_F(p).

Example

Consider the vector-valued function F:R2→R2displaystyle F:mathbb R ^2to mathbb R ^2 defined by:

F(x,y)=[excos⁡yexsin⁡y].displaystyle F(x,y)=beginbmatrixe^xcos y\e^xsin y\endbmatrix.

The Jacobian matrix is:

JF(x,y)=[excos⁡y−exsin⁡yexsin⁡yexcos⁡y]displaystyle J_F(x,y)=beginbmatrixe^xcos y&-e^xsin y\e^xsin y&e^xcos y\endbmatrix

with Jacobian determinant:

detJF(x,y)=e2xcos2⁡y+e2xsin2⁡y=e2x.displaystyle det J_F(x,y)=e^2xcos ^2y+e^2xsin ^2y=e^2x.,!

The determinant e2xdisplaystyle e^2x is nonzero everywhere. Thus the theorem guarantees that, for every point pdisplaystyle p in R2displaystyle mathbb R ^2, there exists a neighborhood about pdisplaystyle p over which Fdisplaystyle F is invertible. This does not mean Fdisplaystyle F is invertible over its entire domain: in this case Fdisplaystyle F is not even injective since it is periodic: F(x,y)=F(x,y+2π)displaystyle F(x,y)=F(x,y+2pi ).

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 (which can also be used as the key step in the proof of existence and uniqueness of solutions to ordinary differential equations).
Since the fixed point theorem applies in infinite-dimensional (Banach space) settings, this proof generalizes immediately to the infinite-dimensional version of the inverse function theorem (see "Generalizations", below).
An alternate proof in finite dimensions hinges on 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: bounds on the derivative of the function imply an estimate of the size of the neighborhood on which the function is invertible.^[2]

Generalizations

Manifolds

The inverse function theorem can be rephrased in terms of differentiable maps between differentiable manifolds. In this context the theorem states that for a differentiable map F:M→Ndisplaystyle F:Mto N (of class C1displaystyle C^1), if the differential of Fdisplaystyle F,

dFp:TpM→TF(p)Ndisplaystyle dF_p:T_pMto T_F(p)N

is a linear isomorphism at a point pdisplaystyle p in Mdisplaystyle M then there exists an open neighborhood Udisplaystyle U of pdisplaystyle p such that

F|U:U→F(U)_U:Uto F(U)

is a diffeomorphism. Note that this implies that Mdisplaystyle M and Ndisplaystyle N must have the same dimension at pdisplaystyle p.
If the derivative of Fdisplaystyle F is an isomorphism at all points pdisplaystyle p in Mdisplaystyle M then the map Fdisplaystyle F is a local diffeomorphism.

Banach spaces

The inverse function theorem can also be generalized to differentiable maps between Banach spaces Xdisplaystyle X and Ydisplaystyle Y. Let Udisplaystyle U be an open neighbourhood of the origin in Xdisplaystyle X and F:U→Ydisplaystyle F:Uto Y a continuously differentiable function, and assume that the derivative dF0:X→Ydisplaystyle dF_0:Xto Y of Fdisplaystyle F at 0 is a bounded linear isomorphism of Xdisplaystyle X onto Ydisplaystyle Y. Then there exists an open neighbourhood Vdisplaystyle V of F(0)displaystyle F(0) in Ydisplaystyle Y and a continuously differentiable map G:V→Xdisplaystyle G:Vto X such that F(G(y))=ydisplaystyle F(G(y))=y for all ydisplaystyle y in Vdisplaystyle V. Moreover, G(y)displaystyle G(y) is the only sufficiently small solution xdisplaystyle x of the equation F(x)=ydisplaystyle 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→Ndisplaystyle F:Mto N has constant rank near a point p∈Mdisplaystyle pin M, then there are open neighborhoods Udisplaystyle U of pdisplaystyle p and Vdisplaystyle V of F(p)displaystyle F(p) and there are diffeomorphisms u:TpM→Udisplaystyle u:T_pMto U and v:TF(p)N→Vdisplaystyle v:T_F(p)Nto V such that F(U)⊆Vdisplaystyle F(U)subseteq V and such that the derivative dFp:TpM→TF(p)Ndisplaystyle dF_p:T_pMto T_F(p)N is equal to v−1∘F∘udisplaystyle v^-1circ Fcirc u. That is, Fdisplaystyle F "looks like" its derivative near pdisplaystyle p. Semicontinuity of the rank function implies that there is an open dense subset of the domain of Fdisplaystyle F on which the derivative has constant rank. Thus the constant rank theorem applies to a generic point of the domain.

When the derivative of Fdisplaystyle F is injective (resp. surjective) at a point pdisplaystyle p, it is also injective (resp. surjective) in a neighborhood of pdisplaystyle p, and hence the rank of Fdisplaystyle F is constant on that neighborhood, and the constant rank theorem applies.

Holomorphic Functions

If a holomorphic function Fdisplaystyle F is defined from an open set Udisplaystyle U of Cndisplaystyle mathbb C ^n into Cndisplaystyle mathbb C ^n, and the Jacobian matrix of complex derivatives is invertible at a point pdisplaystyle p, then Fdisplaystyle F is an invertible function near pdisplaystyle p. This follows immediately from the real multivariable version of the theorem. One can also show that the inverse function is again holomorphic.^[5]

See also

• Implicit function theorem

Notes

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; 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.