With this wonderful theorem, it is possible to give an analytic description of the two different orientations of a rectifiable simple closed curve. The positive orientation is the one for which Greens theorem holds and the other one, called the negative orientation is the one for which

Definition 12.5.1 Let γ be a parametrization of a simple closed curve Γ of finite length. We say this curve is oriented positively if Green’s theorem holds. That is, whenever

A careful examination of the proof also provides a geometric description of the orientation of a simple closed curve Γ. This method is described in the following procedure. For simple situations, it gives an easy to verify description of positive and negative orientation.
Procedure 12.5.2 Let p be a point of Γ with largest second component and let q be a point of smallest second component, respectively the top and bottom of Γ. There exists a horizontal line segment γ^{∗} having left endpoint x and right endpoint y, part of a horizontal line l such that this segment is a component of l∩U_{i} for U_{i} the inside of Γ and of the two simple curves Γ_{p},Γ_{q} each having endpoints x,y, one contains p and the other contains q. Denote these by Γ_{p} and Γ_{q} respectively. The positive orientation takes the points in the following order: y,p,x,q,y. The negative orientation takes the points in the order x,p,y,q,x.
For most situations, one can use the above method to identify whether a simple closed oriented curve is positively oriented.
There are other regions for which Green’s theorem holds besides just the inside and boundary of a simple closed curve. For Γ a simple closed curve and U_{i} its inside, lets refer to U_{i} ∪ Γ as a Jordan region. When you have two non overlapping Jordan regions which intersect in a finite number of simple curves, you can delete the interiors of these simple curves and what results will also be a region for which Green’s theorem holds. This is illustrated in the following picture.
There are two Jordan regions (inside a simple closed curve) here with insides U_{1i} and U_{2i} and these regions intersect in three simple curves. As indicated in the picture, opposite orientations are given to each of these three simple curves. Then the line integrals over these cancel. The area integrals add. Recall the two dimensional area of a bounded variation curve equals 0.
Denote by Γ the curve on the outside of the whole thing and Γ_{1} and Γ_{2} the oriented boundaries of the two holes which result when the curves of intersection are removed, the orientations as shown. Then letting f

as shown in the following picture,
it follows from applying Green’s theorem to both of the Jordan regions,

where ∂U is oriented as indicated in the picture and involves the three oriented curves Γ,Γ_{1},Γ_{2}.
As a particular case of Green’s theorem applied to a simple closed curve, let
 (12.9) 
f
Then some computations show that for

and so

where −C_{r} is the counterclockwise orientation of the small circle. That second integral is routine to evaluate. A parametrization is

Then the integral is

If the orientation of Γ is opposite, then the same reasoning shows that

This shows that if

One can apply Green’s theorem directly in this case. However, if

and if orientation is clockwise, then

This proves the following interesting result.