The circulation of a vector field $\FF$ around a closed loop $C$ is simply the integral $$\Oint \FF\cdot d\rr$$ We will restrict ourselves in this lesson to the $xy$-plane; the circle on the integral sign then implies both that $C$ is closed, and that the loop is oriented in a counterclockwise direction.

Consider a rectangular loop in the $xy$-plane, with sides parallel to the coordinate axes. What is the circulation of $\FF$ around this loop? Consider first the two horizontal edges, on each of which $d\rr=dx\,\ii$, so that \begin{eqnarray*} \Int_{\rm top+bottom} \hskip-5pt\FF \cdot d\rr &=& - \Int_I \FF \bigg|_{\rm top}\kern-3pt\cdot\ii\> dx + \Int_I \FF \bigg|_{\rm bottom}\kern-10pt\cdot\ii\> dx \\ &=& - \Int_I \FF \bigg|_{\rm bottom}^{\rm top}\kern-10pt\cdot\ii\> dx \end{eqnarray*} where $I$ denotes the common domain of $x$ along the top and bottom edges, and where the signs keep track of the counterclockwise orientation.

An argument similar to the one in the previous lesson shows that $$ \FF \bigg|_{\rm bottom}^{\rm top}\kern-10pt\cdot\ii = F_x \bigg|_{\rm bottom}^{\rm top} = \Int_{\rm bottom}^{\raise2pt\hbox{$\scriptstyle{\rm  top}$}} \kern-5pt dF_x = \Int_{\rm bottom}^{\raise2pt\hbox{$\scriptstyle{\rm  top}$}} \kern-4pt \grad F_x \cdot d\rr = \Int_{\rm bottom}^{\raise2pt\hbox{$\scriptstyle{\rm  top}$}} \kern-3pt \Partial{F_x}{y} \,dy $$ so that $$ \Int_{\rm top+bottom} \hskip-7pt\FF \cdot d\rr = - \DInt{\rm inside} \Partial{F_x}{y} \>dy\,dx $$

Repeating this argument for the remaining two sides, being careful with the orientation, leads to a similar term with $x$ and $y$ interchanged and with the opposite sign, so that the circulation around this loop is \begin{equation} \oint\limits_{\rm loop} \FF \cdot d\rr = \Int_{\rm inside} \left( \Partial{F_y}{x} - \Partial{F_x}{y} \right) dA \label{Green} \end{equation} This is Green's Theorem! We say more about the right-hand side in the next lesson.

If $\FF$ is conservative, then integrating $\FF$ is independent of path, so that $$\Oint\FF\cdot d\rr = 0$$ over any simple closed curve $C$. In particular, in two dimensions we have $$F_x\,\ii + F_y\,\jj \hbox{ is conservative} \Longleftrightarrow \Partial{F_y}{x } - \Partial{F_x}{y } = 0 $$ which is often used as a test for conservative vector fields. Green's Theorem states that, even when $\FF$ is not conservative, there is still a relationship between the circulation and the difference of the partial derivatives.

One of the most interesting consequences of Green's Theorem is that it can be used to compute the area of a region as a line integral along the boundary. If $F_x$ and $F_y$ are any functions such that $$\Partial{F_y}{x } - \Partial{F_x}{y } = 1$$ then Green's Theorem says that $$\Lint\FF\cdot d\rr = \Int_{\rm inside} dA = \hbox{area inside $C$}$$ Typical choices are $$A = \Oint x\,dy = - \Oint y\,dx = {1\over2} \Oint x\,dy - y\,dx$$ It is therefore possible to determine the area of a region by walking around its boundary. Geographers use a gadget called a planimeter to do just that, determing the area of, say, Oregon by tracing its outline on a map. (Nowadays, of course, this is done by computer.)

GOALS