positive orientation of the boundary curve C is given by the orientation of S: if you walk with your body aligned with the normal vector to the surface, and keep the surface to your left, then your path C is positively oriented.
The circulation of a velocity field v around C is the integral
It measures the tendency for the fluid (or whatever is moving!) to flow along C.
Stokes's theorem: Let S be an oriented piecewise-smooth surface that
is bounded by a simple, closed, piecewise-smooth boundary curve C with
positive orientation induced by the orientation of S. Let F be a
vector field whose components have continous partials on an open region in
containing S. Then
In the special case where S is flat, lies in the xy-plane, and is oriented in the positive z direction,
which we encountered as a special form of Green's theorem a few sections back.
For a point 
in a fluid with velocity field v,
where 
is the boundary of the disk of radius a and with orientation
given by n. If you like you can think of this a definition of the curl!
It gives us a good way of thinking of ``curl''.
If 
and 
are oriented surfaces with the same boundary curve C, and
both satisfy the hypotheses of Stokes's theorem, then
So if one surface is easy to integrate over while the other is hard, choose to work with the easier!
Stokes's theorem is a useful generalization of Green's theorem, which sheds light on the meaning of the curl of a vector field (which we can think of as measuring the circulation of a vector field - that is, the rotation of the field about axes in the field).
While it is not explicitly derived, the curl of a vector field over a closed surface is zero: what goes curling in must come uncurling out! (see homework exercises 19, p. 1144; and exercise 25, p. 1151, section 17.9).