Inverse Function Theorem

Statements

Theorem. Let $W$ be an open set of ${\mathbb{R}}^n$ and, for a given $r\in {\mathbb{N}}_{>0}$, let $F:W\to {\mathbb{R}}^n$ be a vector-valued function of class ${\mathcal{C}}^r$. If $a\in W$ and the Jacobian Matrix $DF(a)$ is nonsingular, then there exists an open neighborhood $N(a)\subset W$ such that $V=F(N(a))$ is open and $F\in {\mathcal{C}}^r(N(a),V)$ is a diffeomorphism. Morover, for every $x\in N(a)$ and $y=F(x)$, the following formula for the derivatives of $F^{-1}$ $D{F}^{-1}(y)=DF(x{)}^{-1},$ holds, where $D{F}^{-1}$ is the inverse Jacobian matrix.

References

@Boo75 Theorem 6.4