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Green's theorem   (English)

Definition (advanced level)

In two dimensions, if a closed curve C encloses a region R in which functions P(x,y) and Q(x,y) have continuous partial derivatives, then:



C 
( Pdx + Qdy ) =


R 

Q

x
P

y

dA

In three dimensions, if a closed surface S encloses a volume V in which a vector function F has continuous partial derivatives, then:



S 
( F.n ) dA =


V 
( ∇.F ) dV

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