Inverse function theorem: Difference between revisions

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In [[mathematics]], the '''inverse function theorem''' gives sufficient conditions for a [[vector-valued function]] to be [[invertible]] on an [[open region]] containing a point in its ___domain. The theorem can be generalized to maps defined on [[manifold|manifolds]], and on infinite dimensional [[Banach space]]s. Loosely, a ''[[smooth function|C¹]]'' function ''F'' is invertible at a point ''p'' if its [[Jacobian]] ''J_F(p)'' is invertible.

TheMore precisely, the theorem states that if the [[total derivative]] of a [[continuously differentiable]] function ''F'' :defined from an open set U of '''R'''^''n'' →into '''R'''^''n'' is invertible at a point ''p'' (i.e., the [[Jacobian determinant]] of ''F'' at ''p'' is nonzero), andthen ''F'' is [[continuously differentiable]] near ''p'', then it is an invertible function near ''p''. That is, an [[inverse function]] to ''F'' exists in some [[neighbourhood (mathematics)|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 linear map|bounded]] inverse near ''p''.

The Jacobian matrix of ''F''^−-1 at ''F''(''p'') is then the inverse of the Jacobian of ''F'', evaluated at ''p''. This can be understood as a special case of the [[chain rule]], which states that for [[linear transformations]] ''Ff'' and ''Gg'',

:<math>J_{G \circ F} (p) = J_G (F(p)) \cdot J_F (p)</math>

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where J denotes the corresponding Jacobian matrix.

The conclusion of the theorem is that the system of ''n'' equations ''y''_''i'' = ''fF''_''j''(''x''₁,...,''x''_''n'') can be solved for ''x''₁,...,''x''_''n'' in terms of ''y''₁,...,''y''_''n'' if we restrict ''x'' and ''y'' to small enough neighborhoods of ''p''.

Assume that the inverse function theorem holds at ''F''(''p''). Let <math>G(p) = F^{-1}(p)</math>.

:<math>J_{F^{-1} \circ F} (p) = J_{F^{-1}} (F(p)) \cdot J_F (p)</math>

:<math>J_{I} (p) \cdot (J_F (p))^{-1} = J_{F^{-1}} (F(p)) \cdot J_F (p) \cdot (J_F (p))^{-1}</math>

:<math>I \cdot (J_F (p))^{-1} = J_{F^{-1}} (F(p)) \cdot I</math>

:<math>(J_F (p))^{-1} = J_{F^{-1}} (F(p))</math>

where ''I'' is the [[identity transformation]]. This is often expressed more clearly as the useful single-variable formula,

:<math>f'(x) = {{1} \over {(f^{-1})'(f(x))}}.</math>

The conclusion of the theorem is that the system of ''n'' equations ''y''_''i'' = ''f''_''j''(''x''₁,...,''x''_''n'') can be solved for ''x''₁,...,''x''_''n'' in terms of ''y''₁,...,''y''_''n'' if we restrict ''x'' and ''y'' to small enough neighborhoods of ''p''.

The inverse function theorem can be generalized to differentiable maps between [[differentiable manifold]]s. In this context the theorem states that for a differentiable map ''F'' : ''M'' &rarr; ''N'', if the [[pushforward (differential)|derivative]] of ''F'',

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If the derivative of ''F'' is an isomorphism at all points ''p'' in ''M'' then the map ''F'' is a [[local diffeomorphism]].

==ExamplesExample==

Comsider the function ''F'' from '''R'''² to '''R'''² defined by

:<math>

F(x,y)=

\begin{bmatrix}

{e^x \cos y}\\

{e^x \sin y}\\

\end{bmatrix}

</math>

Then the Jacobian matrix is

:<math>

J_F(x,y)=

\begin{bmatrix}

{e^x \cos y} & {-e^x \sin y}\\

{e^x \sin y} & {e^x \cos y}\\

\end{bmatrix}

</math>

and the determinant is

Several functions exist for which differentiating the inverse is much easier than differentiating the function itself. Using the inverse function theorem, a derivative of a function's inverse indicates the derivative of the original function. Perhaps the most well-known example is the method used to compute the derivative of the [[natural logarithm]], whose inverse is the [[exponential function]]. Let <math>u = \ln x</math> and restrict the ___domain to x&nbsp;&gt;&nbsp;0. Then

:<math>

:<math>\frac{d}{dx}\ln x = {{1} \over {\frac{d}{du}e^u}} = {{1} \over {e^u}} = {{1} \over {e^{\ln x}}} = {{1} \over {x}}.</math>

\det J_F(x,y)=

e^{2x} \cos^2 y + e^{2x} \sin^2 y=

e^{2x}.

\,\!</math>

The determinant e^2x is nonzero everywhere. By the theorem, for every point ''p'' in '''R'''², there exists a neighborhood about ''p'' over which ''F'' is invertible.

For more general [[logarithms]], we see that <math>\frac{d}{dx} \log_b(x) = \frac{1}{x \ln(b)} = \frac{\log_b(e)}{x}.</math>

==References==

A similar approach can be used to differentiate an inverse [[trigonometric function]]. Let <math>u = \arctan x.</math> Then

*[[Albert Nijenhuis]]. "Strong derivatives and inverse mappings." ''[[American Mathematical Monthly]]''. Vol. 81, 1974, Pages 969-980.

*[[Walter Rudin]]. ''Principles of Mathematical Analysis''. Third Edition. [[McGraw-Hill]], Inc., 1976, Pages 221-223.

:<math>\frac{d}{dx}\arctan x = {{1} \over {\frac{d}{du}\tan u}} = \cos^2{u} = \cos^2{\arctan x} = \left({{1} \over {\sqrt{1+x^2}}}\right)^2 = {{1} \over {1+x^2}}.</math>

[[Category:Multivariable calculus]]