Green’s Theorem

In this section, we examine Green’s theorem, which is an extension of the Fundamental Theorem of Calculus to two dimensions. Green’s theorem has two forms: a circulation form and a flux form, both of which require region D in the double integral to be simply connected. However, we will extend Green’s theorem to regions that are not simply connected.

Put simply, Green’s theorem relates a line integral around a simply closed plane curve C and a double integral over the region enclosed by C. The theorem is useful because it allows us to translate difficult line integrals into more simple double integrals, or difficult double integrals into more simple line integrals.

Extending the Fundamental Theorem of Calculus

Recall that the Fundamental Theorem of Calculus says that

As a geometric statement, this equation says that the integral over the region below the graph of $F^{\prime }(x)$

and above the line segment $[a,b]$

depends only on the value of F at the endpoints a and b of that segment. Since the numbers a and b are the boundary of the line segment $[a,b],$

the theorem says we can calculate integral ${\displaystyle {\int }_a^{b}F\text{′}(x)dx}$

based on information about the boundary of line segment $[a,b]$

([link]). The same idea is true of the Fundamental Theorem for Line Integrals:

When we have a potential function (an “antiderivative”), we can calculate the line integral based solely on information about the boundary of curve C.

Green’s theorem takes this idea and extends it to calculating double integrals. Green’s theorem says that we can calculate a double integral over region D based solely on information about the boundary of D. Green’s theorem also says we can calculate a line integral over a simple closed curve C based solely on information about the region that C encloses. In particular, Green’s theorem connects a double integral over region D to a line integral around the boundary of D.

Circulation Form of Green’s Theorem

The first form of Green’s theorem that we examine is the circulation form. This form of the theorem relates the vector line integral over a simple, closed plane curve C to a double integral over the region enclosed by C. Therefore, the circulation of a vector field along a simple closed curve can be transformed into a double integral and vice versa.

Notice that Green’s theorem can be used only for a two-dimensional vector field F. If F is a three-dimensional field, then Green’s theorem does not apply. Since

this version of Green’s theorem is sometimes referred to as the tangential form of Green’s theorem.

The proof of Green’s theorem is rather technical, and beyond the scope of this text. Here we examine a proof of the theorem in the special case that D is a rectangle. For now, notice that we can quickly confirm that the theorem is true for the special case in which $\text{F}=\langle P,Q\rangle$

is conservative. In this case,

because the circulation is zero in conservative vector fields. By [link], F satisfies the cross-partial condition, so $P_y=Q_x.$

Therefore,

which confirms Green’s theorem in the case of conservative vector fields.

Proof

Let’s now prove that the circulation form of Green’s theorem is true when the region D is a rectangle. Let D be the rectangle $\left[a,b\right]\times \left[c,d\right]$

oriented counterclockwise. Then, the boundary C of D consists of four piecewise smooth pieces $C_1,$

and $C_4$

([link]). We parameterize each side of D as follows:

Then,

By the Fundamental Theorem of Calculus,

Therefore,

But,

Therefore, ${\displaystyle {\int }_C\text{F}•d\text{r}={\displaystyle \int {\displaystyle {\int }_D\left(Q_x-P_y\right)dA}}}$

and we have proved Green’s theorem in the case of a rectangle.

To prove Green’s theorem over a general region D, we can decompose D into many tiny rectangles and use the proof that the theorem works over rectangles. The details are technical, however, and beyond the scope of this text.

□

In the preceding two examples, the double integral in Green’s theorem was easier to calculate than the line integral, so we used the theorem to calculate the line integral. In the next example, the double integral is more difficult to calculate than the line integral, so we use Green’s theorem to translate a double integral into a line integral.

In [link], we used vector field $\text{F}(x,y)=\langle P,Q\rangle =\langle -\frac{y}{2},\frac{x}{2}\rangle$

to find the area of any ellipse. The logic of the previous example can be extended to derive a formula for the area of any region D. Let D be any region with a boundary that is a simple closed curve C oriented counterclockwise. If $\text{F}(x,y)=\langle P,Q\rangle =\langle -\frac{y}{2},\frac{x}{2}\rangle ,$

then $Q_x-P_y=1.$

Therefore, by the same logic as in [link],

It’s worth noting that if $\text{F}=\langle P,Q\rangle$

is any vector field with $Q_x-P_y=1,$

then the logic of the previous paragraph works. So. [link] is not the only equation that uses a vector field’s mixed partials to get the area of a region.

Flux Form of Green’s Theorem

The circulation form of Green’s theorem relates a double integral over region D to line integral ${\displaystyle {\oint }_C\text{F}·\text{T}ds},$

where C is the boundary of D. The flux form of Green’s theorem relates a double integral over region D to the flux across boundary C. The flux of a fluid across a curve can be difficult to calculate using the flux line integral. This form of Green’s theorem allows us to translate a difficult flux integral into a double integral that is often easier to calculate.

Because this form of Green’s theorem contains unit normal vector N, it is sometimes referred to as the normal form of Green’s theorem.

Proof

Recall that ${\displaystyle {\oint }_C\text{F}·\text{N}ds}={\displaystyle {\oint }_C\text{−}Qdx+Pdy}.$

Let $M=\text{−}Q$

and $N=P.$

By the circulation form of Green’s theorem,

□

Recall that if vector field F is conservative, then F does no work around closed curves—that is, the circulation of F around a closed curve is zero. In fact, if the domain of F is simply connected, then F is conservative if and only if the circulation of F around any closed curve is zero. If we replace “circulation of F” with “flux of F,” then we get a definition of a source-free vector field. The following statements are all equivalent ways of defining a source-free field $\text{F}=\langle P,Q\rangle$

on a simply connected domain (note the similarities with properties of conservative vector fields):

1. The flux ${\displaystyle {\oint }_C\text{F}·\text{N}ds}$

   across any closed curve C is zero.

2. If $C_1$

   and

   $C_2$

   are curves in the domain of F with the same starting points and endpoints, then

   ${\displaystyle {\int }_{C_1}\text{F}·\text{N}ds}={\displaystyle {\int }_{C_2}\text{F}·\text{N}ds}.$

   In other words, flux is independent of path.

3. There is a stream function $g(x,y)$

   for F. A stream function for

   $\text{F}=\langle P,Q\rangle$

   is a function g such that

   $P=g_y$

   and

   $Q=\text{−}{g}_x.$

   Geometrically,

   $\text{F}=\left(a,b\right)$

   is tangential to the level curve of g at

   $\left(a,b\right).$

   Since the gradient of g is perpendicular to the level curve of g at

   $\left(a,b\right),$

   stream function g has the property

   $\text{F}\left(a,b\right)•\text{∇}g\left(a,b\right)=0$

   for any point

   $\left(a,b\right)$

   in the domain of g. (Stream functions play the same role for source-free fields that potential functions play for conservative fields.)

4. $P_x+Q_y=0$

Vector fields that are both conservative and source free are important vector fields. One important feature of conservative and source-free vector fields on a simply connected domain is that any potential function $f$

of such a field satisfies Laplace’s equation $f_{xx}+f_{yy}=0.$

Laplace’s equation is foundational in the field of partial differential equations because it models such phenomena as gravitational and magnetic potentials in space, and the velocity potential of an ideal fluid. A function that satisfies Laplace’s equation is called a harmonic function. Therefore any potential function of a conservative and source-free vector field is harmonic.

To see that any potential function of a conservative and source-free vector field on a simply connected domain is harmonic, let $f$

be such a potential function of vector field $\text{F}=\langle P,Q\rangle .$

Then, $f_x=P$

and $f_x=Q$

because $\text{∇}f=\text{F}.$

Therefore, $f_{xx}=P_x$

and $f_{yy}=Q_y.$

Since F is source free, $f_{xx}+f_{yy}=P_x+Q_y=0,$

and we have that $f$

is harmonic.

Green’s Theorem on General Regions

Green’s theorem, as stated, applies only to regions that are simply connected—that is, Green’s theorem as stated so far cannot handle regions with holes. Here, we extend Green’s theorem so that it does work on regions with finitely many holes ([link]).

Before discussing extensions of Green’s theorem, we need to go over some terminology regarding the boundary of a region. Let D be a region and let C be a component of the boundary of D. We say that C is positively oriented if, as we walk along C in the direction of orientation, region D is always on our left. Therefore, the counterclockwise orientation of the boundary of a disk is a positive orientation, for example. Curve C is negatively oriented if, as we walk along C in the direction of orientation, region D is always on our right. The clockwise orientation of the boundary of a disk is a negative orientation, for example.

Let D be a region with finitely many holes (so that D has finitely many boundary curves), and denote the boundary of D by $\partial D$

([link]). To extend Green’s theorem so it can handle D, we divide region D into two regions, $D_1$

and $D_2$

(with respective boundaries $\partial D_1$

and $\partial D_2),$

in such a way that $D=D_1\cup D_2$

and neither $D_1$

nor $D_2$

has any holes ([link]).

Assume the boundary of D is oriented as in the figure, with the inner holes given a negative orientation and the outer boundary given a positive orientation. The boundary of each simply connected region $D_1$

and $D_2$

is positively oriented. If F is a vector field defined on D, then Green’s theorem says that

Therefore, Green’s theorem still works on a region with holes.

To see how this works in practice, consider annulus D in [link] and suppose that $\text{F}=\langle P,Q\rangle$

is a vector field defined on this annulus. Region D has a hole, so it is not simply connected. Orient the outer circle of the annulus counterclockwise and the inner circle clockwise ([link]) so that, when we divide the region into $D_1$

and $D_2,$

we are able to keep the region on our left as we walk along a path that traverses the boundary. Let $D_1$

be the upper half of the annulus and $D_2$

be the lower half. Neither of these regions has holes, so we have divided D into two simply connected regions.

We label each piece of these new boundaries as $P_i$

for some i, as in [link]. If we begin at P and travel along the oriented boundary, the first segment is $P_1,$

then $P_2,P_3,$

and $P_4.$

Now we have traversed $D_1$

and returned to P. Next, we start at P again and traverse $D_2.$

Since the first piece of the boundary is the same as $P_4$

in $D_1,$

but oriented in the opposite direction, the first piece of $D_2$

is $\text{−}{P}_4.$

Next, we have $P_5,$

then $\text{−}{P}_2,$

and finally $P_6.$

[link] shows a path that traverses the boundary of D. Notice that this path traverses the boundary of region $D_1,$

returns to the starting point, and then traverses the boundary of region $D_2.$

Furthermore, as we walk along the path, the region is always on our left. Notice that this traversal of the $P_i$

paths covers the entire boundary of region D. If we had only traversed one portion of the boundary of D, then we cannot apply Green’s theorem to D.

The boundary of the upper half of the annulus, therefore, is $P_1\cup P_2\cup P_3\cup P_4$

and the boundary of the lower half of the annulus is $\text{−}{P}_4\cup P_5\cup -P_2\cup P_6.$

Then, Green’s theorem implies

Therefore, we arrive at the equation found in Green’s theorem—namely,

The same logic implies that the flux form of Green’s theorem can also be extended to a region with finitely many holes:

Key Concepts

• Green’s theorem relates the integral over a connected region to an integral over the boundary of the region. Green’s theorem is a version of the Fundamental Theorem of Calculus in one higher dimension.
• Green’s Theorem comes in two forms: a circulation form and a flux form. In the circulation form, the integrand is $\text{F}·\text{T}.$

  In the flux form, the integrand is

  $\text{F}·\text{N}.$
• Green’s theorem can be used to transform a difficult line integral into an easier double integral, or to transform a difficult double integral into an easier line integral.
• A vector field is source free if it has a stream function. The flux of a source-free vector field across a closed curve is zero, just as the circulation of a conservative vector field across a closed curve is zero.

Key Equations

• Green’s theorem, circulation form

  ---

  ${\displaystyle {\oint }_{C}Pdx+Qdy={\displaystyle {\iint }_D{Q}_x-P_{y}dA,}}$

  where C is the boundary of D

• Green’s theorem, flux form

  ---

  ${\displaystyle {\oint }_C\text{F}·d\text{r}}={\displaystyle {\iint }_D{Q}_x-P_{y}dA,}$

  where C is the boundary of D

• Green’s theorem, extended version

  ---

  ${\displaystyle {\oint }_{\partial D}\text{F}·d\text{r}}={\displaystyle {\iint }_D{Q}_x-P_{y}dA}$