Diffeomorphism on
If
and are open sets in , we say that is a diffeomorphism if is a bijection and both and are differentiable.
Local Diffeomorphism on
A map
where is a local diffeomorphism about if there exists an open neighbourhood of such that is a diffeomorphism onto its image.
Remark
If we do not specify
, it is infinitely times differentiable (smooth) by default. General definitions of diffeomorphism and local diffeomorphism are given here.
Inverse Function Theorem
A
map where for which is invertible is a local diffeomorphism about .
Implicit Function Theorem
A map
be a smooth map. And such that the first n columns of are linearly independent. Then there exists a neighbourhood