Page "Tangent space" Paragraph 43

: Theorem.

If φ: M → N is a local diffeomorphism at x in M then dφ < sub > x </ sub >: T < sub > x </ sub > M → T < sub > φ ( x )</ sub > N is a linear isomorphism.

Conversely, if dφ < sub > x </ sub > is an isomorphism then there is an open neighborhood U of x such that φ maps U diffeomorphically onto its image.