The curl of this velocity field is
curl V =∇× V =∣∣
∣∣
∣∣ˆe 1 ˆe 2 ˆe 3
∂
∂x∂
∂y∂
∂z
−ω 0 y −ω 0 x 0∣∣
∣∣
∣∣= 2ω^0 ˆe^3.The curl tells us the direction of the angular velocity but not its magnitude.
Stokes Theorem
Let F=F(x, y, z)denote a vector field having continuous derivatives in a region
of space. Let Sdenote an open two-sided surface in the region of the vector field.
For any simple closed curve Clying on the surface S, the following integral relation
holds ∫∫
S(curl F)·dS=∫∫S(
∇× F)
·ˆendS =∫C©F·dr, (8 .39)where the surface integrations are understood to be over the portion of the surface
enclosed by the simple closed curve Clying on Sand the line integral around Cis in
the positive sense with respect to the normal vector to the surface bounded by the
simple closed curve C. The above integral relation is known as Stokes theorem.^1 In
scalar form, the line and surface integrals in Stokes theorem can be expressed as
∫∫S( curl F)·dS=∫∫Scurl F·ˆendS=∫∫S[(
∂F 3
∂y −∂F 2
∂z)
ˆe 1 +(
∂F 1
∂z −∂F 3
∂x)
eˆ 2 +(
∂F 2
∂x −∂F 1
∂y)
ˆe 3]
·ˆendSand
∫C©F·dr =∫C©F 1 dx +F 2 dy +F 3 dz,(8 .40)where eˆnis a unit normal to the surface Sinside the closed curve C. In this case the
path of integration Cis counterclockwise with respect to this normal. By the right-
hand rule if you place the thumb of your right hand in the direction of the normal,
then your fingers indicate the direction of integration in the counterclockwise or
positive sense.
(^1) George Gabriel Stokes (1819-1903) An Irish mathematician who studied hydrodynamics.