text: In potential or irrotational flow, the vorticity vector field is zero, i.e., 
  where  is the velocity field and  is the vorticity field. Like any vector field having zero curl, the velocity field can be expressed as the gradient of certain scalar, say  which is called the velocity potential, since the curl of the gradient is always zero. We therefore have
  In flow regions where vorticity is known to be important, such as wakes and boundary layers, potential flow theory is not able to provide reasonable predictions of the flow. Fortunately, there are often large regions of a flow where the assumption of irrotationality is valid which is why potential flow is used for various applications. For instance in: flow around aircraft, groundwater flow, acoustics, water waves, and electroosmotic flow.
  In potential flow, the circulation  around any simply-connected contour  is zero. This can be shown using the Stokes theorem, 
  where  is the line element on the contour and  is the area element of any surface bounded by the contour. In multiply-connected space (say, around a contour enclosing solid body in two dimensions or around a contour enclosing a torus in three-dimensions) or in the presence of concentrated vortices, (say, in the so-called irrotational vortices or point vortices, or in smoke rings), the circulation  need not be zero. In the former case, Stokes theorem cannot be applied and in the later case,  is non-zero within the region bounded by the contour. Around a contour encircling an infinitely long solid cylinder with which the contour loops  times, we have 
  where  is a cyclic constant. This example belongs to a doubly-connected space. In an - connected space, there are  cyclic constants, namely,