A formula that expresses the connection between the flow of
a vector field through a two-dimensional oriented manifold and the circulation
of this field along the correspondingly oriented boundary of this manifold. Let
be an oriented piecewise-smooth surface, let
be the unit normal to
(at those points, of course, where it exists), which defines the orientation of
,
and let the boundary of
consist of a finite number of piecewise-smooth contours. The boundary of
is denoted by
,
and is oriented by means of the unit tangent vector
,
such that the orientation of
obtained is compatible with the orientation
of
.
If
is a continuously-differentiable
vector field
in a neighbourhood of
,
then
(
is the area element of
,
is the differential of the arc length of the boundary
of
)
or, in coordinate form,
Stated by
G. Stokes
(
1854).