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)
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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)
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If the region is on the left when traveling around , then area of can be computed using the elegant formula
(3)
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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)
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which gives the signed area enclosed by the curve.
The symmetric form above corresponds to Green's theorem with and , leading to
(5)
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(6)
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(7)
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(8)
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(9)
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However, we are also free to choose other values of and , including and , giving the "simpler" form
(10)
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and and , giving
(11)
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A similar procedure can be applied to compute the moment about the -axis using and as
(12)
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and about the -axis using and as
(13)
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where the geometric centroid is given by and .
Finally, the area moments of inertia can be computed using and as
(14)
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using and as
(15)
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and using and as
(16)
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