−∫ dφ = φ1 − φ2,
(2)
so that the flow is independent of the curve for all curves mutually reconcilable; and the circulation round a closed curve is zero, if the curve can be reduced to a point without leaving a region for which φ is single valued.
If through every point of a small closed curve the vortex lines are drawn, a tube is obtained, and the fluid contained is called a vortex filament.
By analogy with the spin of a rigid body, the component spin of the fluid in any plane at a point is defined as the circulation round a small area in the plane enclosing the point, divided by twice the area. For in a rigid body, rotating about Oz with angular velocity ζ, the circulation round a curve in the plane xy is
| ∫ ζ ( x | dy | − y | dx | ) ds = ζ times twice the area. |
| ds | ds |
(3)
In a fluid, the circulation round an elementary area dxdy is equal to
| u dx + ( v + | dv | dx ) dy − ( u + | du | dy ) dx − vdy = ( | dv | − | du | ) dx dy, |
| dx | dy | dx | dy |