The curl F can be expressed by using the operator ∇in the determinant form
curl F =∇× F =∣∣
∣∣
∣∣ˆe 1 ˆe 2 ˆe 3
∂
∂x∂
∂y∂
∂z
F 1 F 2 F 3∣∣
∣∣
∣∣=ˆe^1 [∂F 3
∂y −∂F 2
∂z ]−eˆ^2 [∂F 3
∂x −∂F 1
∂z ] + eˆ^3 [∂F 2
∂x −∂F 1
∂y ] (8.38)Example 8-9. Find the curlof the vector field
F=x^2 yˆe 1 + (x^2 +y^2 z)ˆe 2 + 4 xyz ˆe 3Solution From the relation (8.38) one finds
curl F =∣∣
∣∣
∣∣ˆe 1 ˆe 2 ˆe 3
∂
∂x∂
∂y∂
∂z
x^2 y x^2 +y^2 z 4 xyz∣∣
∣∣
∣∣curl F = (4xz −y^2 )ˆe 1 − 4 yz ˆe 2 + (2 x−x^2 )ˆe 3.Physical Interpretation of Curl
The curlof a vector field is itself a vector field. If curl F = 0 at all points of a
region R, where F is defined, then the vector field F is called an irrotational vector
field , otherwise the vector field is called rotational.
The circulation
∫C©F·dr about a point P 0 can be written as
∫C©F ·dr =∫C©F·drds ds =∫C©F·ˆetdswhere Cis a simple closed curve about the point P 0 enclosing an area. The quantity
F·eˆt, evaluated at a point on the curve C, represents the projection of the vector
F(x, y, z ), onto the unit tangent vector to the curve C. If the summation of these
tangential components around the simple closed curve is positive or negative, then
this indicates that there is a moment about the point P 0 which causes a rotation.
The circulation is a measure of the forces tending to produce a rotation about a given
point P 0 .The curl is the limit of the circulation divided by the area surrounding P 0
as the area tends toward zero. The curl can also be thought of as a measure of the
circulation density of the field or as a measure of the angular velocity produced by
the vector field.