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.

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]] 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''.

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]] ''f'' and ''g'',