Let $D^2$ denote the closed unit disk in $\mathbb{R}^2$. Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$ (and on $D^2$ by restriction). Let $\phi$ be a diffeomorphism of $D^2$ which is equal to the identity in a neighborhood of $\partial D^2$, and which preserves area; i.e. $\phi^* \omega = \omega$. There is a $1$-form $\alpha$ with $d\alpha = \omega$. We have $\phi^*\alpha - \alpha$ is exact, and equal to $df$ for some smooth function $f$ which vanishes on $\partial D$. The Calabi invariant of $\phi$ is the integral$$C(\phi) = \int_{D^2} f\omega.$$What is an example where the Calabi invariant is nontrivial?