Math Help - Help with Stokes Theorem

1. Help with Stokes Theorem

The question states: F(x,y,z)=(y-z, 7z-8, x-7y). Compute line integral F.ds along the unit circle c(t)=(cos(t), sin(t), 0), 0<=t<=2pi.

So by doing the line integral F(c(t)).c'(t) you get -2pi. But how do you go about it using Stokes Theorem? What is the parametrization and what form of it would you use? Curl of F.dS?

2. Originally Posted by arperidot
The question states: F(x,y,z)=(y-z, 7z-8, x-7y). Compute line integral F.ds along the unit circle c(t)=(cos(t), sin(t), 0), 0<=t<=2pi.

So by doing the line integral F(c(t)).c'(t) you get -2pi. But how do you go about it using Stokes Theorem? What is the parametrization and what form of it would you use? Curl of F.dS?
You cannot do this with Stoke's theorem. Because Stoke's theorem is computing surface integrals,

3. Originally Posted by ThePerfectHacker
You cannot do this with Stoke's theorem. Because Stoke's theorem is computing surface integrals,
Actually you can. The surface is $z = 0, x^2+y^2 \le 1$

$
\nabla \times F = < -14,-2, -1>
$

and

$\vec{n} = <0,0,1>$ so $\nabla \times F \cdot \vec{n} = -1$

Thus

$
\iint_R \nabla \times F \cdot \vec{n} dS = -\iint_R 1 dA = -2 \pi
$
noting that since $z = 0, dA = dx dy = r dr d \theta$