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 , 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 alsoArea,
Area Moment of Inertia,
Curl Theorem,
Divergence Theorem,
Geometric Centroid,
Multivariable Calculus,
Stokes' Theorem Explore with Wolfram|Alpha ReferencesArfken, 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. Referenced on Wolfram|AlphaGreen's Theorem Cite this as:Weisstein, Eric W. "Green's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GreensTheorem.html
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