07 - Inverse Function Theorem in ℝ²

Inverse Function Theorem in ${\mathbb{R}}^2$

Let $\Omega \subseteq {\mathbb{R}}^2$ be open, $f\in C^1(\Omega ,{\mathbb{R}}^2)$ and $\underline{a}\in \Omega$
Suppose that $df(\underline{a}):{\mathbb{R}}^2\to {\mathbb{R}}^2$ is invertible then

There exists an open neighbourhood $U$ of $\underline{a}$ such that

$$f(U)\text{ is open }\text{ and }f:U\to f(U)\text{ is a bijection }$$

Where $f$ has an inverse $f^{-1}\in C^1(f(U),U)$
With $d{f}^{-1}(f(\underline{a}))=df(\underline{a}{)}^{-1}$

Showing if a function is locally invertible $f$ be a function of $(x,y)$ Need to find the Jacobian Matrix $df(x,y)$ We need the determinant of the matrix to be positive (and continuous) in the region So that it is invertible and hence the theorem applies

Let