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Problem :

Find the divergence and curl of the vector field F = (2y, 2x + 3z, 3y).

Both are simple calculations. We start with the divergence:

= (2y) + (2x + 3z) + (3y) = 0

Now the curl:


=(3y) - (2x + 3z),(2y) - (3y),(2x + 3z) - (2y)  
 =(0, 0, 0)  

It just so happens that both div and curl are zero.

Problem :

Refer to the last problem. What is the surface integral over any closed surface in the field? What is the line integral over any closed loop?

Stokes' Theorem states that, if the curl of a field is zero, then the line integral over any closed loop is zero, implying that over any closed loop in our field, the line integral is zero. Similarly, Gauss' Theorem states that, if the divergence of a field is zero, then the surface integral over any closed surface is zero. This applies to our field as well.

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