Introduction.
The Fundamental Theorem of Calculus in the form
\begin{equation*}
\int\limits_{[a,b]} \frac{df}{dx} dx = f(b)-f(a)
\end{equation*}
gives the integral of the derivative of a function over an interval in terms of the values of the function itself at the "edges" of that interval.
Green’s Theorem is one of three results that extend this idea to multiple integrals, where the "edge" of a domain becomes a curve for double integrals and a surface for triple integrals, so the values at the edge must be integrated over such curves or surfaces.
Subsection 6.4.1 Simple Closed Curves, Positive Orientation, and Green’s Theorem
The simplest case is for a double integral over a region whose boundary is a simple closed curve, where simple means that the boundary curve does not intersect itself (except that its terminal point is the same as its initial point).
The value of a path integral can depend on the direction of movement along the path, which for a simple closed curve can be described as clockwise or anti-clockwise; to clarify, we specify a default direction of rotation:
Definition 6.4.1. Positive Orientation.
A simple closed curve has positive orientation if it is traversed anti-clockwise (the "trigonometric" direction).
Another way to think of this for a curve \(\C\) that is the boundary of a region \(D\) is that when moving forward along the curve, the interior of the region is to the left.
Alternative Notation.
The use of positive orientation for the integral around a simple closed curve is sometimes indicated by \(\oint_\C\) and the boundary of \(D\) is sometime denoted \(\partial D\text{,}\) giving the alternative notations
\begin{equation*}
\int_\C P dx + Q dy = \oint_\C P dx + Q dy = \oint_{\partial D} P dx + Q dy
\end{equation*}
Theorem 6.4.2. Green’s Theorem (Circulation Form).
Let \(D\) be a domain in the plane whose boundary can be described by a positively oriented simple closed curve \(\C = \partial D\text{.}\)
For \(f(x,y)\) a function that is continuous on \(D\) and on an open region surrounding \(D\text{,}\)
\begin{equation}
\oint_{\partial D} f(x,y)\ dx
= - \iint\limits_D \frac{\partial f}{\partial y}\ dA\tag{6.4.1}
\end{equation}
and
\begin{equation}
\oint_{\partial D} f(x,y)\ dy =
\iint\limits_D \frac{\partial f}{\partial x}\ dA\tag{6.4.2}
\end{equation}
Thus for \(\vec{F} = \vector{P, Q}\) continuous on \(D\) and on an open region surrounding \(D\text{,}\) the circulation around the boundary of \(D\) is
\begin{align}
\oint_{\partial D} \vec{F} \cdot d\vec{r}
= \oint_{\partial D} \vec{F} \cdot \T\ ds
\amp= \oint_{\partial D} P\ dx + Q\ dy\notag\\
\amp= \iint\limits_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\ dA\tag{6.4.3}
\end{align}
Subsection 6.4.3 An Application of Green’s Theorem: When the Cross-Partials Condition Implies That A Vector Field is Conservative
The cross-partials condition for a vector field to be conservative on a simply connected domain,
Theorem 6.3.8, was left unproven in
Section 6.3.
The inference in one direction is already given by
Theorem 6.3.6 so we only need to verify that the cross-partials condition implies that the field is conservative.
We indicate why this is true by arguing that the cross-partials condition ensures that
\(\int_\C \vec{F} \cdot d\vec{r}=0\text{,}\) which was seen in
Theorem 6.3.5 to be equivalent to being conservative.
If a closed path \(\C\) is simple, it surrounds a simply connected domain \(D'\) within the simply connected domain \(D\text{,}\) because if anything inside \(\C\) were not part of \(D\text{,}\) there would be a "hole" in \(D\text{.}\) (This is intuitive, but not a rigorous proof).
Green’s theorem then gives
\begin{equation*}
\int_\C \vec{F} \cdot d\vec{r} = \int_\C P dx + Q dy = \int_{D'} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \ dA = \int_{D'} 0\ dA = 0.
\end{equation*}
Next, for a non-simple path, divide it into pieces that are simple closed curves by cutting at each point where it crosses itself (again this is intuitive but not really proved here). The path integral around \(\C\) is thus zero because it is the sum of the integrals around each of these simple closed curves, and each of those integrals is zero as argued above.
Subsection 6.4.5 Green’s Theorem for Non-simply Connected Domains
If a domain \(D\) has "holes", it can intuitively be considered as coming from a larger domain \(E\) by removing one or more simply connected pieces \(H_1\text{,}\) \(H_2\text{,}\) etc., so that \(E = D \cup H_1 \dots\text{.}\)
Then intuitively
\begin{equation*}
\iint\limits_E \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\ dA
= \iint\limits_D \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\ dA
+ \iint\limits_{H_1} \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\ dA
+ \cdots
\end{equation*}
or
\begin{equation*}
\iint\limits_D \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\ dA
= \iint\limits_E \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\ dA
- \iint\limits_{H_1} \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\ dA
\ \cdots
\end{equation*}
All the integrals at right are over simply connected regions, so Green’s Theorem in the form of Equation
Theorem 6.4.2 allows each to be written as a circulation.
This motivates
Theorem 6.4.7.
For a region \(D\) consisting of an enclosing simply connected region \(E\) minus simply connected "holes" \(H_1\) and so on, so that it has "outer boundary" \(\partial E\) and inner boundaries around each hole,
\begin{equation*}
\iint\limits_D \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\ dA
= \oint\limits_{\partial E} P\ dx + Q\ dy
- \oint\limits_{\partial H_1} P\ dx + Q\ dy
\ \cdots
\end{equation*}
That is, the integral of \(Q_x - P_y\) over region \(D\) is the circulation around the outer boundary curve minus the circulation around each of the inner boundary curves.
Proof.
The more careful proof of this is done by cutting with curves \(\C_1\) and so on from the outer boundary of \(D\) to each of the holes, giving a simply connection region whose boundary consists of
The outer boundary \(\partial E\text{,}\) cut into several pieces.
The boundary of each hole, but traversed clockwise, so that the pieces are \(-\partial H_1\) and so on.
The two sides of each of the cut curves: \(\C_1\text{,}\) \(-C_1\) and so on.
The cuts do not change what is in the region, so the double integral is unchanged. On the other hand, the integrals along the two sides of each cut cancel out, leaving the boundary circulation integral as
\begin{gather*}
\oint_{\partial D} P\ dx + Q\ dy + \oint_{-\partial H_1} P\ dx + Q\ dy\ ...\\
= \oint_{\partial D} P\ dx + Q\ dy - \oint_{\partial H_1} P\ dx + Q\ dy\ ...
\end{gather*}
Because of this, the boundary of such a non-simply connected region is sometimes denoted
\begin{equation*}
\partial D = \partial E - \partial H_1 \dots
\end{equation*}
