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.

==Statement of the theorem==

More 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), then F 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''.

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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'' → ''N'', if the [[pushforward (differential)|derivative]] of ''F'',

:(''DF'')_''p'' : T_''p''''M'' → 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'' → ''F''(''U'')

is a [[diffeomorphism]]. Note that this implies that ''M'' and ''N'' must have the same dimension.

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

==Example==

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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.

==Generalizations==

===Manifolds===

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'' → ''N'', if the [[pushforward (differential)|derivative]] of ''F'',

:(d''DFF'')_''p'' : T_''p''''M'' → 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'' → ''F''(''U'')

is a [[diffeomorphism]]. Note that this implies that ''M'' and ''N'' must have the same dimension.

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 space]]s. 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 (d''F'')₀ : ''X'' → ''Y'' of ''F'' at 0 is a [[bounded linear map|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''.

==References==

*[[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.

* {{cite journal

| last = Nijenhuis

| first = Albert

|authorlink= Albert Nijenhuis

| title = Strong derivatives and inverse mappings

| journal = [[The American Mathematical Monthly|Amer. Math. Monthly]]

| volume = 81

| year = 1974

| pages = 969–980

}}

* {{cite book

| author = Renardy, Michael and Rogers, Robert C.

| title = An introduction to partial differential equations

| series = Texts in Applied Mathematics 13

| edition = Second edition

|publisher = Springer-Verlag

| ___location = New York

| year = 2004

| pages = 337–338

| id = ISBN 0-387-00444-0

}}

* {{cite book

| last = Rudin

| first = Walter

|authorlink= Walter Rudin

| title = Principles of mathematical analysis

| edition = Third edition

| series = International Series in Pure and Applied Mathematics

|publisher = McGraw-Hill Book Co.

| ___location = New York

| year = 1976

| pages = 221–223

}}

[[Category:Multivariable calculus]]