This flow has zero divergence.

 
This vector field has no component out of the screen, so it is only a two-dimensional flow. In the language of cylindrical coordinates, the vector field only has an r-component. If the image is moving too fast, press the stop button on your browser bar.
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    Think about the following questions.
  1. In cylindrical coordinates, on what variables (r, phi, z) does the vector field depend? You should be able to convince yourself that it only depends on r.
  2. Now, notice the area subtended by the red arc and the black arc. This is an area embedded in the flow, so as it moves you can see (sort of) whether the area increases or decreases. And since there is no flow out of the screen, an embedded volume is proportional to the embedded area (make sure you understand this). Can you convince yourself that the embedded area is constant? Looking at the static image may help.
  3. The formula for this vector field is Ar = k/r, where Ar is the radial component of the vector field A. Look up the divergence in cylindrical coordinates and convince yourself that this field has zero divergence.
 
 
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Here is a link to a field with postiive divergence. Click here now.
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