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## Curl

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.

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)

Next: The Laplacian Operator, Up: Curl , , Triple Previous: Curl , , Triple
Prof. Alan Hood
2000-11-06