Past exam of the short options course of the University of Oxford / 2021 / 4 / a / i

Take $\Omega ={\mathbb{R}}^2$ andIts Jacobian matrix and determinant areThus its total derivative is invertible everywhere, and the inverse function theorem makes it locally invertible. But $F(s,t+2\pi )=F(s,t)$, so it is not injective and has no global inverse on its image. The domain is open and connected as required.