Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.7 VECTOR OPERATORS

10.7.3 Curl of a vector field

Thecurlof a vector fielda(x, y, z) is defined by

curla=∇×a=

(

∂az

∂y

−

∂ay

∂z

)

i+

(

∂ax

∂z

−

∂az

∂x

)

j+

(

∂ay

∂x

−

∂ax

∂y

)

k,

whereax,ayandazare thex-,y-andz- components ofa. The RHS can be

written in a more memorable form as a determinant:

∇×a=

∣

∣

∣

∣

∣

∣

∣

∣

ijk

∂

∂x

∂

∂y

∂

∂z

ax ay az

∣

∣

∣

∣

∣

∣

∣

∣

, (10.35)

where it is understood that, on expanding the determinant, the partial derivatives

in the second row act on the components ofain the third row. Clearly,∇×ais

itself a vector field. Any vector fieldafor which∇×a= 0 is said to beirrotational.

Find the curl of the vector fielda=x^2 y^2 z^2 i+y^2 z^2 j+x^2 z^2 k.

The curl ofais given by

∇φ=

∣∣

∣

∣∣

∣

∣∣

ijk

∂

∂x

∂

∂y

∂

∂z

x^2 y^2 z^2 y^2 z^2 x^2 z^2

∣∣

∣

∣∣

∣

∣∣

=− 2

[

y^2 zi+(xz^2 −x^2 y^2 z)j+x^2 yz^2 k

]

.

For a vector fieldv(x, y, z) describing the local velocity at any point in a fluid,

∇×vis a measure of the angular velocity of the fluid in the neighbourhood of

that point. If a small paddle wheel were placed at various points in the fluid then

it would tend to rotate in regions where∇×v= 0 , while it would not rotate in

regions where∇×v= 0.

Another insight into the physical interpretation of the curl operator is gained

by considering the vector fieldvdescribing the velocity at any point in a rigid

body rotating about some axis with angular velocityω.Ifris the position vector

of the point with respect to some origin on the axis of rotation then the velocity

of the point is given byv=ω×r. Without any loss of generality, we may take

ωto lie along thez-axis of our coordinate system, so thatω=ωk. The velocity

field is thenv=−ωyi+ωxj. The curl of this vector field is easily found to be

∇×v=

∣

∣

∣

∣

∣

∣

∣

∣

ijk

∂

∂x

∂

∂y

∂

∂z

−ωy ωx 0

∣

∣

∣

∣

∣

∣

∣

∣

=2ωk=2ω. (10.36)