# 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.