In my investigation of dynamical systems I was met with this seemingly easy question I could not find an answer to:
If we have a two dimensional system of autonomous ODEs viewed as a 2D dynamical system:
$\dot{x} = f(x,y)$
$\dot{y} = g(x,y)$
Is there a simple way to check if this system is area preserving or not, possibly via the Jacobi matrix? I realize this may seem easy and ignorant of me to ask but I just need to know if there is a definitive simple answer. I assume f,g and infinitely differentiable.
The area is preserved if and only if the divergence of the vector field $(f,g)$ is zero everywhere.