Editing Inverse function theorem (section)

== Formulations for manifolds ==
The inverse function theorem can be rephrased in terms of differentiable maps between [[differentiable manifold]]s. In this context the theorem states that for a differentiable map <math>F: M \to N</math> (of class <math>C^1</math>), if the [[pushforward (differential)|differential]] of <math>F</math>,
:<math>dF_p: T_p M \to T_{F(p)} N</math>
is a [[linear isomorphism]] at a point <math>p</math> in <math>M</math> then there exists an open neighborhood <math>U</math> of <math>p</math> such that
:<math>F|_U: U \to F(U)</math>

is a [[diffeomorphism]]. Note that this implies that the connected components of {{Mvar|M}} and {{Mvar|N}} containing ''p'' and ''F''(''p'') have the same dimension, as is already directly implied from the assumption that ''dF''<sub>''p''</sub> is an isomorphism.
If the derivative of {{Mvar|F}} is an isomorphism at all points {{Mvar|p}} in {{Mvar|M}} then the map {{Mvar|F}} is a [[local diffeomorphism]].