In this section we are going to investigate the relationship between certain kinds of line integrals (on closed paths) and double integrals.
Let’s start off with a simple (recall that this means that it doesn’t cross itself) closed curve C and let D be the region enclosed by the curve. Here is a sketch of such a curve and region.
First, notice that because the curve is simple and closed there are no holes in the region D. Also notice that a direction has been put on the curve. We will use the convention here that the curve C has a positive orientation if it is traced out in a counter-clockwise direction. Another way to think of a positive orientation (that will cover much more general curves as well see later) is that as we traverse the path following the positive orientation the region D must always be on the left.
Given curves/regions such as this we have the following theorem.
Let C be a positively oriented, piecewise smooth, simple, closed curve and let D be the region enclosed by the curve. If P and Q have continuous first order partial derivatives onD then,
Before working some examples there are some alternate notations that we need to acknowledge. When working with a line integral in which the path satisfies the condition of Green’s Theorem we will often denote the line integral as,
Both of these notations do assume that C satisfies the conditions of Green’s Theorem so be careful in using them.
Also, sometimes the curve C is not thought of as a separate curve but instead as the boundary of some region D and in these cases you may see C denoted as
Let’s work a couple of examples.
Example 1 Use Green’s Theorem to evaluate
Let’s first sketch C and D for this case to make sure that the conditions of Green’s Theorem are met for C and will need the sketch of D to evaluate the double integral.
So, the curve does satisfy the conditions of Green’s Theorem and we can see that the following inequalities will define the region enclosed.
We can identify P and Q from the line integral. Here they are.
So, using Green’s Theorem the line integral becomes,