For a velocity vector field $\bf{u}$$ = (u,w)$ in two dimensions ($x$ and $z$), we define the stream function $\psi$ to be; $$\psi(P_1) = \int_{P_0}^{P_1} \mathbf{u}\cdot\mathbf{n}dl$$ where $P_0$ is a fixed point and the integral is along a contour from $P_0$ to $P_1$ (conditions are required s.t it doesn't matter which contour we choose). Then for points $P_1$ and $P_2$ it is claimed in various references that $\psi(P_2)-\psi(P_1)$ is the flow rate $\bf{per ~unit~ width}$ between $P_2$ and $P_1$.
My issue is with the unit width part of the statement and I'll give an example to illustrate my problem. In order for the choice of contour not to matter it is shown that we need $$\frac{\partial w}{\partial z} + \frac{\partial u}{\partial x} =0$$ This is fulfilled if we have everywhere a constant vector field. Now choose a point A s.t. the straight line between $P_0$ and A is perpendicular to the constant vector field. Then $$\psi(A)-\psi(P_0) = \psi(A)-0 = \int_{P_0}^{A} \mathbf{u}\cdot\mathbf{n}dl =\int_{P_0}^{A} |\mathbf{u}|dl = |\mathbf{u}|\cdot|\bar{AP_0}|$$
But this would be the total rate of flow between $A$ and $P_0$, not the flow rate per unit width.