TANGENTIAL CASE: ZERO CURL AND OPEN PATH
There are two ways to look at this case.
Either way, Green's theorem
is behind it all.
Here are the two ways:
- (1) If curl(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) All curl free vector--fields are gradient vector fields. That is, curlF = 0 if and only if
F = grad(f) for some function f. (This follows from Green's theorem). Then, by the multivariable chain rule and the fundamental theorem of calculus,
Here (x0,y0) is the starting point, and
(x1,y1) is the starting point.
Now all you have to do is find the function f ...
Either way, the thing to notice in this case is that all that matters is where the path starts, and where it ends, but not how it gets there.