Green's Theorem
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)
|
If the region 
is on the left when traveling around
, then area of 
can be computed using the elegant formula
 |
(3)
|
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]](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/GreensTheorem/Inline7.gif)
,
equation (3) becomes
 |
(4)
|
which gives the signed area enclosed by the curve.
The symmetric for above corresponds to Green's theorem with 
and
, leading to
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
However, we are also free to choose other values of 
and 
, including 
and 
, giving
the "simpler" form
 |
(10)
|
and 
and 
, giving
 |
(11)
|
A similar procedure can be applied to compute the moment about the 
-axis using 
and 
as
 |
(12)
|
and about the 
-axis using 
and 
as
 |
(13)
|
where the geometric centroid 
is
given by 
and 
.
Finally, the area moments of inertia can be computed using 
and 
as
 |
(14)
|
using 
and 
as
 |
(15)
|
and using 
and 
as
 |
(16)
|
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green's theorem
