Green’s theorem can also
be expressed in its “curl form”.
F = P(x,y) i +
Q(x,y) j and dr =
dx i + dy j
So
∫ F . dr =
∫ F . T ds = ∫ Pdx +
Q dy
C C C
where T
is the unit tangential vector to the curve, C, n is the unit normal vector to the
curve and ds is the arc
length along the curve. See the
figure above.
i j k
Now curl
F = ∂/∂x ∂/∂y ∂/∂z =
[∂Q/∂x - ∂P/∂y] k
P(x,y) Q(x,y) 0
So ∫ F . dr = ∫ F . T ds = ∫
∫
curlz F dA (curl form of Green’s Theorem)
C C R
where curlz F
is the z-component of curl F
= curl F . k
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