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 states that if atthe a[[total pointderivative]] ''p''of a function ''fF'' : '''R'''^''n'' → '''R'''^''n'' hasis invertible at a point ''p'' (i.e., the [[Jacobian determinant]] thatof ''F'' at ''p'' is nonzero), and ''fF'' is [[continuously differentiable]] near ''p'', then it is an invertible function near ''p''. That is, an [[inverse function]] to ''fF'' exists in some [[neighbourhood (mathematics)|neighborhood]] of ''fF''(''p'').

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

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

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|derivative]] of ''f'', (''Df'')_''p'' : T_''p''''M'' &rarr; 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'' &rarr; ''f''(''U'') is a [[diffeomorphism]]. Note that this implies that ''M'' and ''N'' must have the same dimension.

where J denotes the corresponding Jacobian matrix.

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

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

This can be expressed more clearly as <math>f'(a) = {{1} \over {(f^{-1})'(f(a))}}</math>. Where ' indicates the derivative of the function.

<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 inverse function theorem can be generalized to differentiable maps between [[differentiable manifold]]s. In this context the theorem states that for a differentiable map ''fF'' : ''M'' &rarr; ''N'', if the [[pushforward|derivative]] of ''fF'', (''DfDF'')_''p'' : T_''p''''M'' &rarr; T_{''fF''(''p'')}''N'' is a linear isomorphism at a point ''p'' in ''M'' then there exists an open neighborhood ''U'' of ''p'' such that ''fF''|_''U'' : ''U'' &rarr; ''fF''(''U'') is a [[diffeomorphism]]. Note that this implies that ''M'' and ''N'' must have the same dimension.

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

===Examples===