Let
be a smooth path of length
in the domain of a flow velocity function
and let
be such that
is the equivalent unit speed parametrization of(so that
).
The circulation ofalongis
whereis evaluated along the contour
The flux ofacross
For each
has magnitude 1 so
for some
then
and
so that
and
If for a velocity function
around 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 of
across the unit circle is
The circulation of
alongis
