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The curl of the vector is
or

(1.21) 
Thus,
represents the amount of curl or twist
in a flow. This is illustrated in the next example.
Example 1. .9Imagine a stick in a flow given by
. How does it rotate? The angle to a
fixed direction can increase in time due to a spatially
varying flow. Figure 1.14 illustrates this.
Figure 1.14:
A flow causes a stick to rotate since the
distance moved by the right hand side exceeds the distance moved by
the left hand side.

The right hand side moves a distance
while the left hand side moves a distance
. Thus
the change in angle is given by
where
Assuming the is small we may approximate the tangent
function by its small argument expansion. Thus,
Using these expressions we may manipulate the equations to get the
rate of which the angle is changing as
Now a similar effect occurs due to the velocity component .
This time a larger value of
than
causes the angle to decrease in time and
Thus, the net effect of both velocity components is t rotate the stick
by an amount proportional to
This is the component of
.
The curl of the velocity is given the special name, vorticity. Thus,
the vorticity, is

(1.22) 
The amount of curl or twist in a magnetic field is given by
and is proportional to the electric current, .

(1.23) 
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Prof. Alan Hood
20001106