Circulation and Flux in a Velocity Field

Letbe a smooth path of lengthin the domain of a flow velocity functionand letbe such thatis the equivalent unit speed parametrization of(so that).

The circulation ofalongiswhereis evaluated along the contour

The flux ofacross

For eachhas magnitude 1 sofor somethenand

so thatand

If for a velocity functionaround every closed curve in the fluid whose motion is described by the velocity field then the velocity field is said to be circulation free.

Example: The flux ofacross the unit circle is

The circulation ofalongis