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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.

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 (x0,y0) 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 Dana-Picard
2001-05-30