Green’s Theorem
gives the relationship between a line integral around a simple closed
curve in a plane, C, and a double integral
over the enclosed plane region R
bounded by C.
∫ P dx + Q dy =
∫ ∫ [∂Q/∂x -
∂P/∂y] dA (standard form of Green’s Theorem)
C R
The curve, C, is said to be positively
oriented when traveling counterclockwise around
C
keeping the region, R, enclosed to the left. If
F(x,y) =
P(x,y) i
+ Q(x,y) j and
dr
= dx
i
+ dy
j, then F . dr = Pdx + Qdy
So
∫ F . dr =
∫ ∫ [∂Q/∂x -
∂P/∂y] dA
C R
Also
∫ F . dr =
∫ F . Tds = ∫ ∫ curlz F
dA
(curl form of Green’s Theorem)
C C R
And
∫ F . n ds = ∫
∫ div
F dA (divergence form of Green’s
Theorem)
C R
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