Inverse function theorem: Difference between revisions

Revision as of 14:52, 18 March 2004

Revision as of 07:30, 10 April 2004

Line 1:

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 states that if at a point ''P'' a function ''f'':'''R'''^''n'' → '''R'''^''n'' has a [[Jacobian]] determinant that is nonzero, and ''F'' is continuously differentiable near ''P'', it is an invertible function near ''P''. That is, an [[inverse function]] exists, in some [[neighborhood]] of ''F(P)''.

The theorem states that if at a point ''P'' a function

:''f'':'''R'''^''n'' → '''R'''^''n''

has a [[Jacobian]] determinant that is nonzero, and ''F'' is continuously differentiable near ''P'', it is an invertible function near ''P''. That is, an [[inverse function]] exists, in some [[neighborhood]] of ''F(P)''.

The Jacobian matrix of ''f''^-1 at ''f''(''P'') is then the inverse of J''f'', evaluated at P.