Multivariable calculus and vector analysis

The idea behind Green’s theorem

The idea

When $C$ is an oriented closed path (i.e., a path where the endpoint is the same as the beginning point), the integral

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}& & \hfill \end{array}$$

represents the circulation of $\mathbf{F}$ around $C$. If $\mathbf{F}$ were the velocity field of water flow, for example, this integral would indicate how much the water tends to circulate around the path in the direction of its orientation.

One way to compute this circulation is, of course, to compute the line integral directly. But, if our line integral happens to be in two dimensions (i.e., $\mathbf{F}$ is a two-dimensional vector field and $C$ is a closed path that lives in the plane), then Green’s theorem applies and we can use Green’s theorem as an alternative way to calculate the line integral.

Green’s theorem transforms the line integral around $C$ into a double integral over the region inside $C$. However, it’s not obvious what function we should integrate over the region inside $C$ so that we still get the same answer as the line integral. To figure out what we should integrate, the notion of circulation is quite helpful.

Think of the integral ${\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}$ as the “macroscopic” circulation of the vector field $\mathbf{F}$ around the path $C$. Now, imagine you came up with a “microscopic” version of circulation around a curve. This microscopic circulation at a point $\left(x,y\right)$ has to tell you how much $\mathbf{F}$ would circulate around a tiny closed curve centered around $\left(x,y\right)$. We could picture the microscopic circulation as a bunch of small closed curves (shown below in green), where each curve respresents the tendency for the vector field to circulate at that location (imagine that the small curves were really, really small, much smaller than pictured).

Green’s theorem is simply a relationship between the macroscopic circulation around the curve $C$ and the sum of all the microscopic circulation that is inside $C$. If $C$ is a closed curve in the plane (remember, we are talking about two dimensions), then it surrounds some region $D$ (shown in red) in the plane. $D$ is the “interior” of the curve $C$.

Green’s theorem says that if you add up all the microscopic circulation inside $C$ (i.e., the microscopic circulation in $D$), then that total is exactly the same as the macroscopic circulation around $C$.

“Adding up” the microscopic circulation in $D$ means taking the double integral of the microscopic circulation over $D$. Therefore, we can write Green’s theorem as

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}={\iint }_D\text{ “microscopic circulation of }\mathbf{F}\text{” }dA.& & \hfill \end{array}$$

What is this microscopic circulation? We encountered microscropic circulation with the curl of a three-dimensional vector field. The microscopic circulation of Green’s theorem is the same thing. The only difference is that Green’s theorem applies only with two-dimensional vector fields, e.g., for vector fields in the $xy$-plane. The microscropic circulation we want is circulation in the $xy$-plane.

Microscopic circulation parallel to the $xy$ plane turns out to be the $z$-component of the curl. You can see this as follows. The direction of the curl and the definition of its components is determined by the right-hand rule. (Imagine curling the fingers of your right hand around the circles indicating the circulation. One represents such circulation by a vector pointing in the direction of your right thumb.) In three-dimensions, point the thumb of your right hand in the positive $z$ direction. Then, the fingers of your right hand curl in the counterclockwise direction parallel to the $xy$-plane. So, the right-hand rule says that circulation in the $xy$-plane should correspond to the $z$-component of the curl.

Hence, for Green’s theorem,

$$\begin{array}{lll}\hfill \text{ “microscopic circulation”}=\left(\mathrm{curl}\mathbf{F}\right)\cdot \mathbf{k},& & \hfill \end{array}$$

(where $\mathbf{k}$ is the unit-vector in the $z$-direction) and we can write Green’s theorem as

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}={\iint }_D\left(\mathrm{curl}\mathbf{F}\right)\cdot \mathbf{k}dA.& & \hfill \end{array}$$

If you look at the formula at the end of the reading on the components of the curl, you’ll see that the third component of the curl is

$$\begin{array}{lll}\hfill \left(\mathrm{curl}\mathbf{F}\right)\cdot \mathbf{k}=\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y},& & \hfill \end{array}$$

so we can write Green’s theorem as

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}={\iint }_D\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA.& & \hfill \end{array}$$

Note how the right hand rule says that $\left(\mathrm{curl}\mathbf{F}\right)\cdot \mathbf{k}$ corresponds to the amount of circulation in the counterclockwise direction. Hence, Green’s theorem, as we have written it, is valid only for curves oriented counterclockwise (as pictured above). In this case, we say that $C$ is a positively oriented boundary of the region $D$. One way to remember what positively oriented means is the following: if you were to walk along $C$ in the positive orientation, the region $D$ will be to your left. If you mess up the orientation, you’ll be off by a minus sign.

You can read more about some alternative notations for Green’s theorem. Note that Green’s theorem applies more generally that indicated here; it also applies to regions with holes.

When Green’s theorem applies

Green’s theorem provides another way to calculate

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}& & \hfill \end{array}$$

that you can use instead of calculating the line integral directly. However, Green’s theorem works only for the case where $C$ is a closed curve. If $C$ is an open curve (i.e., a curve where the endpoint isn’t the same as the beginning point), please don’t even think about using Green’s theorem.

If you think of the above idea, you won’t make this mistake. Green’s theorem converts the line integral to a double integral of the microscopic circulation. The double integral is taken over the region $D$ inside the path. Only closed paths have a region $D$ inside them. The idea of circulation makes sense only for closed paths.

So if you are asked to compute the integral

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_{C}ydx+xydy& & \hfill \end{array}$$

where $C$ is the line from $\left(0,1\right)$ to $\left(1,1\right)$, can you use Green’s theorem? No, because $C$ is an open curve.

You could, for example, use Green’s theorem to compute

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_{C}ydx+xydy& & \hfill \end{array}$$

where $C$ is the counterclockwise unit circle.

One other thing: Green’s theorem can be used only for vector fields in two dimensions, such as the $\mathbf{F}\left(x,y\right)=\left(y,xy\right)$ of the above example. It cannot be used for vector fields in three dimensions. So, don’t bother with Green’s theorem if you are given an integral like

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_{C}zdx+xydy-yzdz& & \hfill \end{array}$$

even if $C$ is a closed path.

Why am I going on and on about this? Because students frequently under pressure try to use Green’s theorem when it doesn’t apply. I hope that if you’ve read this, you’ll resist that temptation.

You can read Green’s theorem examples here.

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

5a. Green's theorem

• The idea behind Green's theorem
• Other ways of writing Green's theorem
• Green's theorem with multiple boundary components
• Using Green's theorem to find area
• Sketch of proof for circulation per unit area
• Green's theorem examples

5b. The theorem for conservative vector fields

5c. Stokes' theorem

5d. The divergence theorem

6. Review material