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]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FGreensTheorem%2FInline7.svg){kind=small}
, equation (3) becomes
 |
(4) |
which gives the signed area enclosed by the curve.
The symmetric form above corresponds to Green's theorem with 
and 
, leading to
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) |
See also
Area, Area Moment of Inertia, Curl Theorem, Divergence Theorem, Geometric Centroid, Multivariable Calculus, Stokes' Theorem
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References
Arfken, G. "Gauss's Theorem." §1.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985.Kaplan, W. "Green's Theorem." §5.5 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 286-291, 1991.
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Cite this as:
Weisstein, Eric W. "Green's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GreensTheorem.html