Inverse function theorem: Difference between revisions

Revision as of 05:42, 2 December 2023

Revision as of 10:26, 27 December 2023

Line 20:

In other words, if <math>Jf^{-1}(b), Jf(a)</math> are the Jacobian matrices representing <math>(f^{-1})'(b), f'(a)</math>, this means:

:<math>Jf^{-1}(b) = Jf(a)^{-1}.</math>

The hard part of the theorem is the existence and differentiability of <math>f^{-1}</math>. Assuming this, the inverse derivative formula follows from the [[chain rule]] applied to <math>f^{-1}\circ f = I</math>. (Indeed, <math>I = (f^{-1} \circ f)^'(a) = (f^{-1})'(b) \circ f'(a).</math>) Since taking the inverse is infinitely differentiable, the formula for the derivative of the inverse shows that if <math>f</math> is continuously <math>k</math> times differentiable, with invertible derivative at the point {{Mvar|a}}, then the inverse is also continuously <math>k</math> times differentiable. Here <math>k</math> is a positive integer or <math>\infty</math>.

There are two variants of the inverse function theorem.<ref name="Hörmander" /> Given a continuously differentiable map <math>f : U \to \mathbb{R}^m</math>, the first is