1. ## Stokes Theorem

Am studying stoke's theorem and when doin practice problems have little or no trouble with one exception,

When i have a Surface such as the cone z=(x^2 + y^2)^1/2 with z<= 2

And i parameterize the curve ds c(t) = (2cost , 2sint, 2) it always seems to have the wrong sign, as in answers parameterization is;

c(t) = (2cost, - 2sint, 2).

I'm probably missing something obvious but with all my study brain is fried.
Any help would be greatly appreciated.

Cheers.

2. Originally Posted by monster
Am studying stoke's theorem and when doin practice problems have little or no trouble with one exception,

When i have a Surface such as the cone z=(x^2 + y^2)^1/2 with z<= 2

And i parameterize the curve ds c(t) = (2cost , 2sint, 2) it always seems to have the wrong sign, as in answers parameterization is;

c(t) = (2cost, - 2sint, 2).

I'm probably missing something obvious but with all my study brain is fried.
Any help would be greatly appreciated.

Cheers.
they both parametrize the curve but in different directions when the curve is seen from the above. the point is that in Stokes' theorem you can use both directions but you have to be careful

with choosing the normal vector $\displaystyle \bold{n}.$ the direction of your parametrization with respect to $\displaystyle \bold{n}$ must follow the right-hand rule. so, for example, in your question, if you choose the parametrization

$\displaystyle C=(2\cos t, 2 \sin t, 2),$ then $\displaystyle \bold{n}=\bold{k}$ but if you choose $\displaystyle C=(2 \cos t, -2 \sin t, 2),$ then $\displaystyle \bold{n}=-\bold{k}.$