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Curl of a plane vector field.
Let
be a vector field in the plane, i.e. at every point (x,y) in the plane we attach a vector
.
We will define the circulation density of
at a point in the plane.
Consider a small rectangle, whose sides are parallel to the coordinate axes, as in
Figure 8.
Figure 8:

On each side of the rectangle, we have the following flow rates:
Top side 

Right side 

Bottom 

Left side 

Combine opposite sides:
Top and Bottom 

Right and left sides 

Add the two formulas on the right, divide by the area of the rectangle (
), then take the limit when
and
approach 0. We get the circulation density of
at the point P(x,y).
Definition 6.30
The
circulation density or
curl of the vector field
at the point (
x_{0},
y_{0}) is the number
Example 6.31
Let
.
Then
and
.
We have:
.
Next: The divergence of a
Up: Applications of the derivative.
Previous: Lagrange's multipliers.
Noah DanaPicard
20010530