Inverse Function Theorem (multivariable)

Inverse Function Theorem: Let $U\subseteq\mathbb{R}^n$ be open and let $f:U\to\mathbb{R}^n$ be of class $C^1$. Suppose $a\in U$ and $\det Df(a)\neq 0.$ Let $b=f(a)$. Then there exist open neighborhoods $U_0$ of $a$ and $V_0$ of $b$ such that:

• $f:U_0\to V_0$ is bijective ,
• the inverse $f^{-1}:V_0\to U_0$ is of class $C^1$, and
• the derivative of the inverse is given by $Df^{-1}(b) = (Df(a))^{-1},$ and more generally $Df^{-1}(y)=(Df(f^{-1}(y)))^{-1}$ for $y\in V_0$.

This theorem is the rigorous foundation for local coordinate changes and for solving $f(x)=y$ locally when the linearization is invertible. See also diffeomorphism .

Proof sketch: Since $Df(a)$ is invertible, $f$ is well-approximated near $a$ by the invertible linear map $Df(a)$. Using the mean value inequality , one shows that a suitable Newton-type map is a contraction on a small ball (after composing with $(Df(a))^{-1}$), giving existence and uniqueness of local solutions via the contraction mapping principle. Differentiability of the inverse follows from differentiating the identity $f(f^{-1}(y))=y$.