The Curl of a Vector Field
Let F =F(x, y, z) = F 1 (x, y, z)ˆe 1 +F 2 (x, y, x )ˆe 2 +F 3 (x, y, z)ˆe 3 denote a continuous
vector field possessing continuous derivatives, and let P 0 denote a point in this vector
field having coordinates (x 0 , y 0 , z 0 ). Insert into this field an arbitrary surface Swhich
contains the point P 0 and construct a unit normal ˆento the surface at point P 0 .On
the surface construct a simple closed curve Cwhich encircles the point P 0 .The work
done in moving around the closed curve is called the circulation at point P 0. The
circulation is a scalar quantity and is expressed as
∫
C
©F·dr =Circulation of F around Con the surface S,
where the integration is taken counterclockwise. If the circulation is divided by
the area ∆S enclosed by the simple closed curve C, then the limit of the ratio
Circulation
Area as the area ∆Stends toward zero, is called the component of the curlof
F in the direction ˆen and is written as
( curl F)·ˆen= lim∆S→ 0
∫
C
© F·dr
∆S. (8 .34)
To evaluate one component of the curlof a vector field F at the point P 0 (x 0 , y 0 , z 0 ),
construct the plane z=z 0 which passes through P 0 and is parallel to the xy plane.
This plane has the unit normal eˆn =ˆe 3 at all points on the plane. In this plane,
consider the circulation at P 0 due to a circle of radius centered at P 0 .The equation
of this circle in parametric form is
x=x 0 +cos θ, y =y 0 +sin θ, z =z 0
and in vector form r = (x 0 +cos θ)ˆe 1 + (y 0 +sin θ)ˆe 2 +z 0 ˆe 3 .The circulation can be
expressed as
I=
∫
C
©F·dr =
∫ 2 π
0
F(x 0 +cos θ, y 0 +sin θ, z 0 ) [−sin θˆe 1 +cos θˆe 2 ]dθ.
By expanding F=F(x 0 +cos θ, y 0 +sin θ, z 0 )in a Taylor series about = 0,one finds
F(x 0 +cos θ, y 0 +sin θ, z 0 ) = F(x 0 , y 0 , z 0 ) + dF
d
+
2
2!
d^2 F
d^2
+···,
where all the derivatives are evaluated at = 0. The circulation can be written as
I=
∫
C
©F·dr =μ 0 +^2 μ 1 +^3 μ 2 +···,
