Inverse function theorem: Difference between revisions

Revision as of 14:45, 21 July 2006

Revision as of 17:31, 10 October 2006

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

===Examples===

Several functions exist for which differentiating the inverse is much easier than differentiating the function itself. Using the inverse function theorem, a derivative of a function's inverse indicates the derivative of the original function. Perhaps the most well-known example is the method used to compute the derivative of the [[natural logarithm]], whose inverse is the [[exponential function]]. Let <math>u = \ln x</math> and restrict the ___domain to x&nbsp;&gt;&nbsp;0. Then