Inverse function theorem: Difference between revisions

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{{Calculus}}

In [[mathematics]], the '''inverse function theorem''' gives a sufficient condition foris a [[differentiable functiontheorem]] tothat beasserts that, if a [[Invertiblereal function|invertible]] in''f'' has a [[Neighbourhoodcontinuously (mathematics)differentiable function|neighborhoodcontinuous derivative]] ofnear a point inwhere its [[___domainderivative ofis anonzero, function|___domain]]:then, near namelythis point, that its''f'' has an [[derivativeinverse function]]. The inverse function is also [[continuousdifferentiable function|continuousdifferentiable]], and non-zero at the point.''[[inverse Thefunction theoremrule]]'' alsoexpresses gives a formula for theits derivative ofas the resulting[[multiplicative inverse]] function,of calledthe [[inversederivative functionof rule]]''f''.

The theorem applies verbatim to [[complex-valued function]]s of a [[complex number|complex variable]]. It generalizes to functions from

In [[multivariable calculus]], this theorem can be generalized to any [[continuously differentiable]], [[vector-valued function]] whose [[Jacobian determinant]] is nonzero at a point in its ___domain, giving a formula for the [[Jacobian matrix]] of the inverse. There are also versions of the inverse function theorem for [[holomorphic function]]s, for differentiable maps between [[manifold]]s, for differentiable functions between [[Banach space]]s, and so forth.

''n''-[[tuples]] (of real or complex numbers) to ''n''-tuples, and to functions between [[vector space]]s of the same finite dimension, by replacing "derivative" with [[Jacobian matrix]] and "nonzero derivative" with "nonzero [[Jacobian determinant]]".

InIf [[multivariablethe calculus]],function thisof the theorem canbelongs beto generalizeda to anyhigher [[continuouslydifferentiability differentiableclass]], [[vector-valuedthe function]] whose [[Jacobian determinant]]same is nonzero at a point in its ___domain, giving a formulatrue for the [[Jacobianinverse matrix]] of the inversefunction. There are also versions of the inverse function theorem for [[holomorphic function]]s, for differentiable maps between [[manifold]]s, for differentiable functions between [[Banach space]]s, and so forth.

The theorem was first established by [[Émile Picard|Picard]] and [[Édouard Goursat|Goursat]] using an iterative scheme: the basic idea is to prove a [[fixed point theorem]] using the [[contraction mapping theorem]].