Green's theorem
gives the relationship between a line integral around a simple closed
curve, C, in the x-y plane. Stokes’ Theorem extends this
concept to a closed curve in
x-y-z space using the “curl form” of Green’s Theorem as follows:
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3-D Green's
Theorem extended to Stokes’ Theorem
gives :
∫ F (x,y,z) . dr = ∫
∫
curl F . n dS (Stokes’ Theorem)
C S
where F(x,y,z) = P(x,y,z)
i + Q(x,y,z) j +
R(x,y,z) k
Also ∫ ∫ curl F
. n dS =
∫ ∫ curl F
. (ru
x rv)
dA
(Conversion to a surface integral)
S R
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