TANGENTIAL CASE: NON-ZERO CURL, OPEN PATH

TANGENTIAL CASE: NON-ZERO CURL AND OPEN PATH

Here you are almost forced to parameterize, and do things by hand. Green's theroem doesn't say as much in this case.

But still, Green's theorem says something.

Here's what it says:

You don't have path independence, but Green's theroem tells you how the value of the line integral changes when the path is changed.

Suppose C is a given open path, and that D is another simpler path between the same initial and final points. The the path C - D is closed, so Green's theorem can be applied. Let Omega be the enclosed region. Then:

So you can "trade in" the path you've got on a straight line path from start to finish -- but not for free. You must pay a "price" of

and it might well be very easy to compute this. So the price might be very good.