Proving the Positivity of a Curve-Related Integral: Hint Requested

[Ocifer]

Homework Statement

C is a positively orientated simple closed curve
Show that the the following integral is always positive.

$\int P dx + Q dy$

Homework Equations

The Attempt at a Solution

I am actually given a particular P and Q, but I would just like a hint on how to proceed.

The integral is presented in such a way that I am tempted to use Green's Theorem, but I'm not sure if that would make it any better. Basically, I am able to show it no problem if I pick particular curves C, and parameterize them. Can anyone give me a hint as to how to do this in the general case when I don't know C exactly.

 

[mighty2000]

To show that the integral is always positive, you can use Green's Theorem. Since C is a positively orientated simple closed curve, you can use the fact that the area enclosed by C is always positive. Then, using Green's Theorem, you can rewrite the integral as a double integral over the region enclosed by C. This region will always have a positive area, so the integrand will always be positive. Therefore, the integral itself will always be positive. I hope this helps!