Quick question on Stokes theorem

[MarkSA]

After looking over the section in my calculus book on Stokes theorem, I am left very confused. The examples seem to use a different trick to solve each problem.

In one, the formula used is: S F(r(t)) * r'(t)
In another, it is: SS curl F * n ds.
In others still, they apparently use: S F * <1dx,1dy,1dz> (which reminds me of green's theorem?)
And finally, just plain: SS curl F * dS

Are all of these equivalent? I just can't figure out why they are choosing to use one over another in the worked examples. Is there one single method above of using Stokes that can always be used and is easiest to handle?

 

[Unco]

(1) In one, the formula used is: S F(r(t)) * r'(t)
(2) In another, it is: SS curl F * n ds.
(3) In others still, they apparently use: S F * <1dx,1dy,1dz> (which reminds me of green's theorem?)
(4) And finally, just plain: SS curl F * dS

Ignore Stoke's theorem for the moment. (1) and (3) are the same kind of expression; the former is just parameterised. And (2) and (4), where the dS is a vector, are the same by the definition of a surface integral, which I am sure you will find in your notes.