NORMAL CASE: ZERO DIVERGENCE AND OPEN PATH

Either way, the divergence theorem

is behind it all.

Here are the two ways:

• (1) If div(F) = 0, then the only aspects of the path that figure in the answer are the start and finish. So you can "trade in" the path you've got on a straight line path from start to finish. If the original path is very complicated, this can be much easier.

• (2) (This is a bit fancier). All divergence free vector--fields are "perpendicular" gradient vector fields. That is, div(F) = 0 if and only if F = perp(grad(f)) for some function f. (This follows from the divergence theorem). Then, by the multivariable chain rule and the fundamental theorem of calculus,
  

  Here, it is assumed that as you travel from (x,y), the normal vector in question is always to your left. (Doesn't sound right, does it?) If this is not the case: switch the start and finish points, and then it will be!

  For more on this method (especially if that sounded wrong!), see the section of notes in which the perp operation is used to realate normal and tangential linte integrals. Now all you have to do is find the function f ...