Given this very simple sympletic vector space: $\left(\mathbb{R}^2, \mathrm{~d} x \wedge \mathrm{d} y\right)$, how can we show that an area preserving diffeomorphism $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ defines a symplectomorphism?
I believe an area-preserving diffeomorphism is a diffeomorphism that preserves the Lebesgue measure (please correct me if this is not the case or if there is a better definition in the context of symplectic geometry), but given that is the case, how can we relate preserving Lebesgue measure to showing that $\mathrm{~d} x \wedge \mathrm{d} y(u,v)=\mathrm{~d} x \wedge \mathrm{d} y (f(u),f(v))$ which is the definition of symplectomorphism where f is our area-preserving diffeomorphism.