Theorem 8.20 (Flux-Divergence form)
Let
be a vector field over a region
in the plane. Denote by
a simple loop in
,
and by
the region enclosed by this loop.
The outward flux of
across
is equal to the double integral of
over
:
Theorem 8.22 (Circulation-Curl form)
Let
be a vector field over a region
in the plane. Denote by
a simple loop in
,
and by
the region enclosed by this loop.
The counterclockwise circulation of
along
is equal to the double integral of
over
: