MAIN CONCEPTS
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Assume that D can be described in two ways. First, D has an upper portion described by the graph  and a lower portion described by the graph . Then D consist of all points  with

as shown below.

Second, D can also be described in another way with a left portion of the graph  and a right portion of the graph . Then D consist of all  with

as shown below.

We use these descriptions to show explicitly that

and

We get Green’s theorem by adding these equations. Let us first look at the first equation.

From our first description of D, we can think of C comprising of  with x varying from a to b and  with x varying from b to a, maintaining the counter clockwise orientation. Note here that direction, and in turn limits of the integral, is important.

The strategy here is to evaluate the left hand and right hand side of the equation separately, using properties of the line integral for piecewise smooth curves, and show that they lead to a similar equation.

Now that both the left hand and right hand side of the equation simplifies to the same expression, less the minus sign, we conclude that

Using the other description of D and a similar argument we can arrive to the equation

Finally, adding these two equations yields Green’s Theorem.

Here are some of my comments. I emphasize again that students new to this theorem should not attempt to find a geometrical link between the close loop integral and the double integral. Instead, one should only see the theorem solely for the purpose of calculating a tedious close loop integral by transforming it to the double integral.

The proof underlines one important aspect with line integrals and that is direction plays an important role. When we say positively orientated, we place importance on the direction the curve is traveling. Only by considering the direction could we change the limits of the integral and get the intended result.