So there is a particular Calculus III problem regarding a piece-wise function that has been bothering me a lot lately. G is a moving particle that goes from a1 (-1,0) to a2 (1,1) to a3 (1,-1) to a4 (2,0) and then back to a1 (1,1) again.
So we can see that this curve makes two nice, same-sized triangles which are connected. Now, we know that Green's Theorem can be used to find the work of a curve that is simple, closed. But, we see that this curve crosses itself. Does this mean that this has transformed to a ‘complex curve’ and Green's Theorem can no longer be applied here?
However, I was wondering if we can treat it as a union of two different simple curves ‘triangle A’ (clockwise, hence -ve) and ‘triangle B’ (anti-clockwise, hence +ve) and use Green's Theorem to find the work done here by a force field [ F = {sinπ(x-y), cosπ(x+y)}] in this moving particle along the curve G? Then use the basic properties of double integrals to break it up and evaluate it?