Fluids will invariably swirl around, and it is often useful to characterize
the amount of swirling. This is done by introducing the circulation 
, defined as
with the line integral taken is around a closed circuit C in a counter-clockwise sense. By using Stokes' theorem we may write this as a surface integral,
where the vorticity 
of the fluid
is defined as
If the vorticity of the fluid is zero, we say that the fluid is irrotational :
We've already mentioned that there are some useful analogies between
magnetostatics and incompressible fluid flow. I've collected these together
in Table 2.1 below. To transform the magnetostatics expressions
into those for fluid flow, we make the substitutions 
, 
, and 
(where here 
is the mass density of the fluid). If a fluid is
both incompressible and irrotational, then 
and 
, which are the same as the equations of
electrostatics and magnetostatics in free space (i.e., in the absence of
charges and currents).
| Magnetostatics | Incompressible fluid flow | |
 | | |
 (Ampére's Law) |  (circulation) | |
 (energy density of the magnetic field) |  (kinetic energy density of a fluid) |
As an example of fluid flow with vorticity, we can consider a single vortex , which would be produced when draining a bathtub, and which is illustrated in Fig. 2.5.
Figure 2.5: Velocity field of a vortex.
Flow lines are drawn for equal increments of the speed, as shown by arrows. Click here to see animations.
there is only one component of
the velocity field, 
. In calculating the circulation, the line
element 
, so that 
. If the circulation is independent of the
integration path, then we must have 
, with C a constant.
The circulation is then
so that 
. Therefore, the velocity field of a vortex is
By making the substitutions suggested above, we can use this result to find the magnetic field produced by a wire carrying a current I:
