Nonlinear shear map
= Nonlinear shear map
{title2=$(u,v)\mapsto(u+f(v),v)$}
This triangular planar map has diagonal <Jacobian matrix> entries one and <determinant> one, for <differentiable> $f$. Its inverse subtracts $f(v)$, so it is an <area-preserving map>. The analogous vertical map $(u,v)\mapsto(u,v+g(u))$ has the same property. Their composition preserves area even when the shifts are nonlinear, as in the logarithmic <area-preserving seasonal predator-prey map>.