On which regions can Green's theorem NOT be applied?

Question

In my calculus book (Stewart), the theorem is proved for a simple region (I understand that this is being enclosed by a simple curve). But then it is specified that the theorem can be extended for a finite union of simple regions that do not overlap, and even for regions that are not simply connected (that is, with holes).

So it doesn't occur to me about what kind of region the theorem might not apply to.

Answer 1

Green's theorem works only for the case where C is a simple closed curve. If C is an open curve, please don't even think about using Green's theorem.

So what is a simple curve? A curve that does not cross itself. So if the region is a finite union of simple regions that overlaps, the curves that enclose the region will not be simple as they will cross each other. So Green's theorem is not applicable there.

Now comes the question. When can we use Green's theorem?

i) When the curve is simple closed curve (failing any one of the conditions can make damage).

ii)Green's theorem can be used only for vector fields in two dimensions,i.e in $ F(x,y)$ form. It cannot be used for vector fields in three dimensions. So, don't bother with Green's theorem if you are given an integral like $ \int_C Adx+Bdy−Cdz\;$ even if $ C$ is a closed path.