Question

When does Green's Theorem in the Plane fail?

Answer 1

Doesn't the region have to be closed? I don't quite remember.

Answer 2

Yes, the region has to be enclosed, but for also some function F(x,y) = P(x,y)i + Q(x,y) has to be continuous at every point.

Answer 3

can you elaborate on the part

Answer 4

Remember to have a derivative a function most be continuous at every point along the curve.

Since, green's theorem takes the shape of

\[\int\limits_{?}^{?}\int\limits_{?}^{?} (dg/dx - df/dy)da\]

(The derivatives being partials)

Some function P and Q must be continuous.

Answer 5

so if you have \[f=\frac{ 1 }{ x^2+y^2 }\ \left(\begin{matrix}-y \\ x\end{matrix}\right)\] this is not continuous because there is an asymptote at the origin?

Answer 6

Yes at (0,0) the function is not continuous.

Answer 7

Remember it has to due with limits essentially a function is not continuous if

1. lim
x→a f(x) does not exist, or
2. lim
x→a f(x) exists, but ≠ f(a)

Answer 8

I hope this helps :)

Answer 9

I see great! thanks!
now is there a formal way of saying that green's theorem in the plane does not hold true for f. or do we just simply say that it is not continuos at the origin.

Answer 10

Well I suppose if you used one of the two conditions I stated above you could show using limits that the function F is not continuous at (0,0), as such greens theorem cannot apply.

Answer 11

great thanks!

Answer 12

Your welcome :)