# Using Stoke's Theorem to determine plane area enclosed by simple closed plane curve

1. Nov 30, 2012

### moonkey

1. The problem statement, all variables and given/known data
Let C be a simple closed plane curve in space. Let n = ai+bj+ck be a unit vector normal to the plane of C and let the direction on C match that of n. Prove that

(1/2)∫[(bz-cy)dx+(cx-az)dy+(ay-bx)dz]

equals the plane area enclosed by C.

What does the integral reduce to when C is in the xy-plane?

2. Relevant equations

Stoke's Theorem

F.ds=∫(∇×F).dS

3. The attempt at a solution

I really have no idea where to start. Any help would be much appreciated.

2. Nov 30, 2012

### LCKurtz

Re: Using Stoke's Theorem to determine plane area enclosed by simple closed plane cur

Why don't you start by calculating $\nabla \times \vec F$ and plug it in the right side of Stokes' Theorem?

3. Nov 30, 2012

### moonkey

Re: Using Stoke's Theorem to determine plane area enclosed by simple closed plane cur

I don't know what F is

4. Nov 30, 2012

### LCKurtz

Re: Using Stoke's Theorem to determine plane area enclosed by simple closed plane cur

You are given a line integral, written in the form $\oint \vec F\cdot d\vec R$. Can't you pick $\vec F$ out of that?

5. Nov 30, 2012

### moonkey

Re: Using Stoke's Theorem to determine plane area enclosed by simple closed plane cur

I think I might have been reading the question incorrectly (well hopefully). I feel like an idiot. I'll give it a go and hopefully it works out. Thanks LCKurtz

6. Nov 30, 2012

### moonkey

Re: Using Stoke's Theorem to determine plane area enclosed by simple closed plane cur

Got it out

Thanks again for your help LCKurtz