Inverse function theorem

Statement

Let $f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ be a differentiable map such that the determinant of the Jacobian matrix of $f$ at a particular point $p\in \mathbb {R} ^{n}$, is nonzero (i.e. the Jacobian matrix is nonsingular). Then, there exist open neighbourhoods $U\ni p$ and $V\ni f(p)$ such that the restriction of $f$ to $U$ is a diffeomorphism from $p$ to $f(p)$.

Applications

• Implicit function theorem
• Local immersion theorem
• Local submersion theorem