Editing Inverse function theorem (section)

===Constant rank theorem===
The inverse function theorem (and the [[implicit function theorem]]) can be seen as a special case of the constant rank theorem, which states that a smooth map with constant [[rank (differential topology)|rank]] near a point can be put in a particular normal form near that point.<ref name="boothby">{{cite book |first=William M. |last=Boothby |title=An Introduction to Differentiable Manifolds and Riemannian Geometry |url=https://archive.org/details/introductiontodi0000boot |url-access=registration |edition=Second |year=1986 |publisher=Academic Press |location=Orlando |isbn=0-12-116052-1 |pages=[https://archive.org/details/introductiontodi0000boot/page/46 46–50] }}</ref> Specifically, if <math>F:M\to N</math> has constant rank near a point <math>p\in M\!</math>, then there are open neighborhoods {{Mvar|U}} of {{Mvar|p}} and {{Mvar|V}} of <math>F(p)\!</math> and there are diffeomorphisms <math>u:T_pM\to U\!</math> and <math>v:T_{F(p)}N\to V\!</math> such that <math>F(U)\subseteq V\!</math> and such that the derivative <math>dF_p:T_pM\to T_{F(p)}N\!</math> is equal to <math>v^{-1}\circ F\circ u\!</math>. That is, {{Mvar|F}} "looks like" its derivative near {{Mvar|p}}.  The set of points <math>p\in M</math> such that the rank is constant in a neighborhood of <math>p</math> is an open dense subset of {{Mvar|M}}; this is a consequence of [[semicontinuity]] of the rank function. Thus the constant rank theorem applies to a generic point of the domain.

When the derivative of {{Mvar|F}} is injective (resp. surjective) at a point {{Mvar|p}}, it is also injective (resp. surjective) in a neighborhood of {{Mvar|p}}, and hence the rank of {{Mvar|F}} is constant on that neighborhood, and the constant rank theorem applies.