I am following the book "A Mathematical Introduction to Fluid Mechanics" by Chorin y Marsden. The paragraph says it.
"... $u, v$ are the components of $\mathsf{u}$. Assume that the flow is contained in some plane domain $D$ with a fixed boundary $\partial D$, with the boundary condition $u \cdot n = 0$ on $\partial D$, where n is the unit outward normal to $\partial D$. Let us assume $D$ is simply connected (i.e., has no “holes”). Then, by incompressibility, $\partial_x u = -\partial_y v$, and so from vector calculus there is a scalar function $\psi(x, y, t)$ on $D$ unique up to an additive constant such that $u = \partial_y \psi$ and $v = -\partial _x \psi$"
I think the property from calculus the text is referring to is the Gauss-theorem, but really I don't understand how it is applied. Could someone please help me understand what is going on?
If a vector valued function $F$ satisfies $$ \nabla\cdot F=0 $$ or is incompressible in fluid mechanics terminology, on a simply connected domain, then there is a potential function for $F$, here called $\psi$ (really a class of such functions since differentiating forgets constants) whose curl is $F$.
This is some iteration of the more general theorem that closed differential forms are exact if your domain is nice (in a topological sense). The curl of a vector valued function is a 2 form, divergence is a three form.