Proving Stoke's Theorem for a Plane Curve

[moonkey]

Homework Statement

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?

Homework Equations

Stoke's Theorem

∫F.ds=∫(∇×F).dS

The Attempt at a Solution

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

 

[LCKurtz]

Homework Statement

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?

Homework Equations

Stoke's Theorem

∫F.ds=∫(∇×F).dS

The Attempt at a Solution

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

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

 

[moonkey]

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

I don't know what F is

 

[LCKurtz]

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?

 

[moonkey]

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?

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

 

[moonkey]

Got it out

Thanks again for your help LCKurtz