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 , 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)

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.