Vector Fields with no Flow and no Flux

Example 1:  

Here is a picture of F (the arrows are scaled, to make the picture clearer).

Now look at the level curves of the two components of F1. These components are  P1=,  Q1=-2xy.
Their level curves are supposed to be perpendicular to each other. We first show P1, then Q1, then both together.

You can see the curves are perpendicular.

Example 2:   Start with f(x)=Cos[x]. It turns out that  Cos[x-iy]=Cos[x]Cosh[y]+iSin[x]Sinh[y],
so       P3=Cos[x]Cosh[y],      Q3=Sin[x]Sinh[y], and   F3=(Cos[x]Cosh[y],Sin[x]Sinh[y]).
Here is a picture of F3:

Now F3=∇(Sin[x]Cosh[y]), and  the  functions
P4=Sin[x]Cosh[y]    Q4=-Cos[x]Sinh[y]
satisfy the Cauchy-Riemann equations, so, as with F1, the level curves of Q4 are the flow lines of F3.