= Solution
For a two-dimensional <incompressible flow> $\mathbf u=(u,v)$, the continuity equation is $u_x+v_y=0$. Locally, or globally on a <simply connected> fluid region, this permits a <streamfunction> with
$$
u=\psi_y,\qquad v=-\psi_x.
$$
Indeed, the one-form $-v\,dx+u\,dy$ is closed by incompressibility and hence is locally the differential of $\psi$. Conversely these expressions automatically give zero <divergence> by equality of mixed derivatives.
A <streamline> at a fixed time is tangent to the instantaneous <velocity field>. Along it,
$$
\frac{d\psi}{ds}=\nabla\psi\cdot\mathbf u
=\psi_x\psi_y-\psi_y\psi_x=0.
$$
Thus \b[streamlines are connected curves on which the streamfunction is constant]; wherever $\nabla\psi\ne0$, the <level curves> have exactly the required tangent direction. At a <stagnation point>, the tangent is degenerate and distinct branches of a level set need not form a single streamline. For unsteady flow this argument fixes time: it does not identify <streamlines> with particle <pathlines>.
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