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