Green's theorem is a vector identity which is equivalent to the curl theorem in the plane. Over a region 
in the plane with boundary 
, Green's theorem states
 |
(1)
|
where the left side is a line integral and the right side is a surface integral. This can also be written compactly in vector form as
 |
(2)
|
An equivalent flux or normal form is
 |
(3)
|
or, for the vector field 
,
 |
(4)
|
where 
is the outward unit normal vector and 
is the arc length element.
Both forms assume that 
is bounded and that 
and 
have continuous partial derivatives on an open neighborhood
of 
and its boundary.
The boundary is piecewise smooth and positively oriented
so that 
is on the left as it is traversed. The flux form follows from the circulation form
by replacing 
by 
and 
by 
,
and is the two-dimensional divergence theorem
(Poonen 2021, pp. 141-142).
If the region 
is on the left when traveling around 
, then area of 
can be computed using the elegant formula
 |
(5)
|
giving a surprising connection between the area of a region and the line integral around its boundary.
For a plane curve specified parametrically as 
for ![t in [t_0,t_1]](/images/equations/GreensTheorem/Inline19.svg)
, equation (5) becomes
 |
(6)
|
which gives the signed area enclosed by the curve.
The symmetric form above corresponds to Green's theorem with 
and 
, leading to
 |  |  |
(7)
|
 |  |  | |
 |  |  | |
 |  |  | |
 |  |  |
However, we are also free to choose other values of 
and 
, including 
and 
, giving the "simpler" form
 |
(8)
|
and 
and 
,
giving
 |
(9)
|
A similar procedure can be applied to compute the moment about the 
-axis using 
and 
as
 |
(10)
|
and about the 
-axis using 
and 
as
 |
(11)
|
where the geometric centroid 
is given by 
and 
.
Finally, the area moments of inertia can be computed using 
and 
as
 |
(12)
|
using 
and 
as
 |
(13)
|
and using 
and 
as
 |
(14)
|
