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 manifolds, and on infinite dimensional [[Banach space]]s.

The theorem states that if the [[total derivative]] of a function ''F'' : '''R'''^''n'' → '''R'''^''n'' is invertible at a point ''p'' (i.e., the [[Jacobian determinant]] of ''F'' at ''p'' is nonzero), and ''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''). 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'',