A simple closed curve is positively oriented if it is traversed in a counterclockwise direction. This means that the region is to the left as we traverse C.
Let C be a positively oriented, piecewise-smooth, simple closed curve in the
plane and let D be the region bounded by C (C is sometimes denoted
in this case, as the boundary of D). If P and Q have
continuous partial derivatives on an open region that contains D, then
Pardon the re-use of the 
symbol in this way! The symbol 
indicates that the boundary is traversed in the positively oriented way.
From 17.3 (the fundamental theorem of line integrals) we discovered that for
gradient (or conservative) fields 
Hence
which means that Green's theorem in the case of gradient fields becomes
We knew that!
Green's theorem can be used in a sneaky way to calculate areas: if we find functions P and Q so that
then we can use a line integral to compute area! Fortunately such pairs of P and Q are easy to find:
- P=0, Q=x;
- P=-y, Q=0;
- P=-y/2, Q=x/2
Green's theorem is simply a calculation of a rather special integral on a two-dimensional region D:
It shows one way to handle the ``work problem'' 
when the field is not conservative. It can also be seen as a
generalization of the Fundamental Theorem of Calculus to area integrals, in the
sense that the integral defined on a region can be evaluated by considering
only its boundary.
We can use this backwards, however: if the boundary calculation is ugly, it may be that the area integral is easier!
Let's show how Green's theorem is just a generalization of the Fundamental Theorem of Calculus to area integrals: consider
for a simple region D. If we can think of a P(x,y) such that
then
Now let's consider
where 
and 
are the integrals over the bottom and top of C,
respectively. Using 
as the parameterization for and similarly
for , we have
