where
μ 0 =∫ 2 π0F 0 ·dξ, F 0 =F(x 0 , y 0 , z 0 )μ 1 =∫ 2 π0dF
d ·d
ξμ 2 =∫ 2 π01
2!d^2 F
d^2·dξ···where all the derivatives are evaluated at = 0 and dξ= (−sin θˆe 1 + cos θˆe 2 )dθ. The
vector F 0 is a constant and the integral μ 0 is easily shown to be zero. The vector
dF
devaluated at = 0 , when expanded is given by
dF
d =∂F
∂x cos θ+∂F
∂y sin θ=(
∂F 1
∂x cos θ+∂F 1
∂y sin θ)
ˆe 1+(
∂F 2
∂x cos θ+∂F 2
∂y sin θ)
ˆe 2+(
∂F 3
∂xcos θ+∂F^3
∂ysin θ)
ˆe 3 ,where the partial derivatives are all evaluated at = 0.It is readily verified that the
integral μ 1 reduces to
μ 1 =π(
∂F 2
∂x −∂F 1
∂y)
.The area of the circle surrounding P 0 is π^2 ,and consequently the ratio of the circu-
lation divided by the area in the limit as tends toward zero produces
( curl F)·ˆe 3 =∂F^2
∂x−∂F^1
∂y. (8 .35)
Similarly, by considering other planes through the point P 0 which are parallel to the
xz and yz planes, arguments similar to those above produce the relations
( curl F)·ˆe 2 =∂F 1
∂z −∂F 3∂x and ( curl
F)·ˆe 1 =∂F^3
∂y −∂F 2
∂z. (8 .36)Adding these components gives the mathematical expression for curl F. One finds
the curl F can be written as
curl F =(
∂F 3
∂y−∂F^2
∂z)
ˆe 1 +(
∂F 1
∂z−∂F^3
∂x)
ˆe 2 +(
∂F 2
∂x−∂F^1
∂y)
ˆe 3. (8 .37)