Vector Calculus-Field Lines vs Streamlines I am an engineer and I have a question related to vector calculus. I find hard to visualise how the field lines of the velocity vector differ from streamlines. I am given that $\bf{V=ui+vj}$.
Also I am given than $\bf{u = \partial ψ/ \partial y}$ and $\bf{v = -\partial ψ/ \partial x.}$ (supposed to be partial derivatives) $\bf{[1]}$
Given the above, it can be shown that for constant $\bf{ψ , \ dψ}$ is equal to 0, hence by using $\bf{dψ=(\partial ψ/ \partial x)dx + (\partial ψ/ \partial y)dy}$ and $\bf{[1],}$
I get that $\bf{dy/dx = v/u. \ [2]}$
What is the physical significance of $\bf{[2]}$? I think it shows that $\bf{ψ}$ corresponds to streamline's equation, but I am not sure "why?".
Also, once I am given the equation of the velocity of the flow, how can I make the corresponding stream line plot?
 A: If I understand your terminology right, the velocity field lines in the steady flow (the lines such that at each point of the line the corresponding velocity vector is tangent to the line) are exactly the same as streamlines (the lines over which a small inertialess particle carried by the flow would travel), so there is no difference except for the name.
As to the second question, what your computation really shows for that particular choice of the velocity field is that
$$
(d\psi)(V)=\frac{\partial\psi}{\partial x}u+\frac{\partial\psi}{\partial y}v=
\frac{\partial\psi}{\partial x}\frac{\partial\psi}{\partial y}+\frac{\partial\psi}{\partial y}\left(-\frac{\partial\psi}{\partial x}\right)=0\,
$$
so $\psi$ is constant along the field lines of $V$, or, if you prefer, along the streamlines, which means that the streamlines are level lines of $\psi$ (lines given by the equation $\psi=const$). Level lines of a known function are usually easier to compute, visualize, and investigate than the field lines of an arbitrary velocity field, so knowing that $\psi$ when it exists really does give you some advantage.
