Two Forms for Green’s Theorem
The notation 
for integration around a closed curve. We elaborate further on the notation here.
A simple closed curve C can be traversed in two possible directions. (Recall that a curve is simple if it does not cross itself.) The curve is traversed counterclockwise, and said to be positively oriented, if the region it encloses is always to the left of an object as it moves along the path. Otherwise it is traversed clockwise and negatively oriented. The line integral of a vector field F along C reverses sign if we change the orientation. We use the notation
for the line integral when the simple closed curve C is traversed counterclockwise, with its positive orientation.
In one form, Green’s Theorem says that the counterclockwise circulation of a vector field around a simple closed curve is the double integral of the k-component of the curl of the field over the region enclosed by the curve.
THEOREM 1 Green’s Theorem (Circulation-Curl or Tangential Form) Let C
be a piecewise smooth, simple closed curve enclosing a region R in the plane. Let 
be a vector field with M and N having continuous first partial derivatives in an open region containing R. Then the counterclockwise circulation of F around C equals the double integral of 
over R.
A second form of Green’s Theorem says that the outward flux of a vector field across a simple closed curve in the plane equals the double integral of the divergence of the field over the region enclosed by the curve.
THEOREM 2 Green’s Theorem (Flux-Divergence or Normal Form) Let C
be a piecewise smooth, simple closed curve enclosing a region R in the plane. Let be a vector field with M and N having continuous first partial derivatives in an open region containing R. Then the outward flux of F across C equals the double integral of div F over the region R enclosed by C.
The two forms of Green’s Theorem are equivalent.
EXAMPLE 3: Verify both forms of Green’s Theorem for the vector field
and the region R bounded by the unit circle
Evaluating 
and computing the partial derivatives of the components of F
The two sides of Equation (4) are
