Verifying Stokes' theorem (orientation?)

[Feodalherren]

Homework Statement

F= <y,z,x>
S is the hemisphere x^2 + y^2 + z^2 = 1, y ≥ 0, oriented in the direction of the positive y-axis.
Verify Stokes' theorem.

Homework Equations

The Attempt at a Solution

So I completed the surface integral part. I'm trying to do the line integral part of Stokes' theorem and end up with the same answer.
Where I get confused is there parametrization part.

I said that r(t) = <cos t, 0, sin t>, 0≤t≤2∏.
Apparently that's the wrong orientation. But when I "grab" the y-axis with my thumb in the positive y-direction and curl my fingers they go from the z axis to the x-axis counter clockwise. Isn't that the CORRECT orientation?
I guess what I'm asking is how do I determine the orientation when I'm using Stokes' theorem. I assume I want the same counter clockwise orientation that I do for Green's theorem.

 

[vela]

Your parameterization goes in the opposite direction. As t goes from 0 to pi/2, r(t) goes from <1, 0, 0> to <0, 0, 1> — in other words, from the x-axis to the z-axis.