Diffeomorphism

A diffeomorphism is a bijection $f:U\to V$ between open sets $U,V\subseteq \mathbb{R}^n$ such that $f$ is differentiable on $U$ and the inverse map $f^{-1}:V\to U$ is differentiable on $V$.

A diffeomorphism is, in particular, a homeomorphism . Sufficient conditions for a map to be locally (and sometimes globally) a diffeomorphism are given by the inverse function theorem , typically phrased using a nonzero Jacobian determinant .

Examples:

• The translation $f(x)=x+a$ on $\mathbb{R}^n$ is a diffeomorphism with inverse $x\mapsto x-a$.
• Any invertible linear map $f(x)=Ax$ (with $\det A\ne 0$) is a diffeomorphism of $\mathbb{R}^n$.