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# The divergence of a vector field in the plane.

Let be a vector field in the plane, i.e. at every point (x,y) in the plane we attach a vector . For example, think of the vector as the velocity vector at the point P(x,y) of a liquid flowing in the plane.

Consider a small rectangle, whose sides are parallel to the coordinate axes, as in Figure  9. In fact the figure is identical to Fig,  8, but we use it differently.

The rate at which the liquid leaves the rectangle through each side of the rectangle is given as follows:

 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 flux density of at the point P(x,y).

Definition 6.32   The flux density or divergence of the vector field at the point (x0,y0) is the number

Example 6.33   Let . Then and . We have: .

Remark 6.34   Suppose that a liquid is entering a region D in the plane through the point P0(x0,y0). The lines of flow diverge at that point and is positive. If the liquid leaves D at the point P0, then is negative.

As an actual example of a vector field, see Figure  10;it has been released on the Internet by the U.S. National Center for Environmental Prediction (NCEP). (http://grads.iges.org/pix/hemi.jet.html).

We quote here part of their comments (200mb Winds):

• Purple shading indicates the speed of the winds at the 200 millibar level, in meters per second. This altitude is near the level of the core of the jet stream. So the tracks of the jet streams can be seen very clearly.
• The streamlines indicate the direction of flow of the wind, which is generally from west to east throughout most of the subtropics, mid- and high-latitudes.
• The color of the streamlines indicates a relative measure of divergence of the flow in the upper troposphere. Orange and red indicates strong divergence at upper levels, usually associated with strong vertical velocities in the middle troposphere, and severe weather/heavy rainfall.

Next: The curl of a Up: Applications of the derivative. Previous: Curl of a plane
Noah Dana-Picard
2001-05-30