Homework Help: Green's Theorem and zero

1. Aug 3, 2010

EV33

1. The problem statement, all variables and given/known data
Use Green's Theorem to evaluate the line integralalone the given positvely oriented curve.

∫$$_{c}$$ sin(y)dx+xcos(y)dy, C is the ellipse x2+xy+y2=4

2. Relevant equations

3. The attempt at a solution
∫∫(cos(y)-cos(y))dA=∫∫0dA

Because this ends up being the double integral of zero, does this just mean my answer is zero?

2. Aug 3, 2010

Dick

That's right. The line integral around any closed curve will be 0.

3. Aug 3, 2010

vela

Staff Emeritus
Yup.

4. Aug 3, 2010

EV33

Awesome. Thank you both.

5. Aug 3, 2010

gomunkul51

Correction: The line integral around any closed curve will be 0 ONLY if the integral is on a conservative field. otherwise it won't be zero for all cases.

6. Aug 3, 2010

Dick

Mmm. Well, sure. F=(sin(y),x*cos(y)) is conservative. So the integral of F.dr is zero around any closed curve. I didn't mean ANY F. Did that really need a 'correction'?

7. Aug 4, 2010

gomunkul51

It's a subtle point, I wanted EV33 to know that :)
It is very easy to make that mistake and think that any close line integral is zero.