Page 385 - Advanced engineering mathematics

11.5 Divergence and Curl 365

11.5.2 A Physical Interpretation of Curl
The curl vector is interpreted as a measure of rotation or swirl about a point. In British literature,
the curl is often called the rot (for rotation) of a vector field.
To understand this interpretation, suppose an object rotates with uniform angular speed ω
about a line L, as in Figure 11.11. The angular velocity vector has magnitude ω and is directed
along L as a right-handed screw would progress if given the same sense of rotation as the object.
Put L through the origin and let R = xi + yj + zk for any point (x, y, z) on the rotating object.
Let T(x, y, z) be the tangential linear velocity and R = R . Then

T = ωR sin(θ) = × R ,

with θ the angle between R and . Since T and × R have the same direction and magnitude,
we conclude that T = × R. Now write = ai + bj + ck to obtain

T = × R = (bz − cy)i + (cx − az)j + (ay − bx)k.

Then

i j k

∇× T = ∂/∂z ∂/∂y ∂/∂z

bz − cy cx − az ay − bx
= 2ai + 2bj + 2ck = 2 .

Therefore,

1
= ∇× T.
2

The angular momentum of a uniformly rotating body is a constant times the curl of the linear
velocity.

(x, y, z)
T
Ω

R sin(θ)
R
θ

(0, 0, 0)
L
FIGURE 11.11 Interpretation of curl.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 15, 2010 16:11 THM/NEIL Page-365 27410_11_ch11_p343-366