Green's Theorem in the Plane
Introduction
If 
is a conservative field, then we know 
for a differentiable function 
and we can calculate the line integral of over any path 
joining point 
to 
. In this section we derive a method for computing a work or flux integral over a closed curve in the plane when the field is not conservative. This method comes from Green’s Theorem, which allows us to convert the line integral into a double integral over the region enclosed by .
Suppose that 
is the velocity field of a fluid flowing in the plane and that the first partial derivatives of 
and 
are continuous at each point of a region 
.