Fluid-dynamics. We have a 2D flow, incompressible. We need to show that $$\exists \psi(x,y,t)$$ such that $$u = \frac{\partial \psi}{\partial y} ,\qquad v = -\frac{\partial \psi}{\partial x}$$
explanation in the book
2D flow $\implies$ $q = (u,v,0)$ . Incompressible $\implies$ $\nabla \cdot q = 0 = \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}$
$\textit{hence, since we are in 2D, there must exists $\psi(x,y,t)$ such that}$ $$u = \frac{\partial \psi}{\partial y}, \qquad v = -\frac{\partial \psi}{\partial x}$$
Why can we conclude that there must exists a $\psi$ like that?
The identity $\partial_x u +\partial_yv = 0$ implies that a vector $(-v,u,0)$ is curl-free. Therefore, by Poincare's second lemma, this vector is a gradient.