6.6 Integration around closed curves, and green's theorem  (Page 5/5)

We need one final lemma before we can complete the general proof of Green's Theorem. This one is where the analysis shows up; there are carefullychosen $ϵ$ 's and $\delta$ 's.

Suppose $S$ is contained in an open set $U$ and that $\omega$ is smooth on all of $U.$ Then Green's Theorem is true.

Let the piecewise smooth geometric set $S$ be determined by the interval $[a,b]$ and the two bounding functions $u$ and $l.$ Using [link] , choose an $r>0$ such that the neighborhood $N_r\left(S\right)\subseteq U.$ Now let $ϵ>0$ be given, and choose $delta$ to satisfy the following conditions:

1. $\delta <r/2,$ from which it follows that the open neighborhood $N_{\delta }\left(S\right)$ is a subset of the compact set ${\overline{N}}_{r/2}\left(S\right).$ (See part (f) of [link] .)
2. $\delta <ϵ/4M,$ where $M$ is a common bound for all four continuous functions $\left|P\right|,\left|Q\right|,|P_y|,$ and $|Q_x|$ on the compact set ${\overline{N}}_{r/2}\left(S\right).$
3. $\delta <ϵ/4M(b-a).$
4. $\delta$ satisfies the conditions of [link] .

Next, using [link] , choose two piecewise linear functions $p_u$ and $p_l$ so that

1.   $|u\left(x\right)-p_u\left(x\right)|<\delta /2$ for all $x\in [a,b].$
2.   $|l\left(x\right)-p_l\left(x\right)|<\delta /2$ for all $x\in [a,b].$
3.   ${\int }_a^b|u^{\text{'}}\left(x\right)-p_u^{\text{'}}\left(x\right)|dx<\delta .$
4.   ${\int }_a^b|l^{\text{'}}\left(x\right)-p_l^{\text{'}}\left(x\right)|dx<\delta .$

Let $\widehat{S}$ be the geometric set determined by the interval $[a,b]$ and the two bounding functions $\widehat{u}$ and $\widehat{l},$ where $\widehat{u}=p_u+\delta /2$ and $\widehat{l}=p_l-\delta /2.$ We know that both $\widehat{u}$ and $\widehat{l}$ are piecewise linear functions. We have to be a bit careful here, since for some $x$ 's it could be that $p_u\left(x\right)<p_l\left(x\right).$ Hence, we could not simply use $p_u$ and $p_l$ themselves as bounding functions for $\widehat{S}.$ We do know from (1) and (2) that $u\left(x\right)<\widehat{u}\left(x\right)$ and $l\left(x\right)>\widehat{l}\left(x\right),$ which implies that the geometric set $S$ is contained in the geometric set $\widehat{S}.$ Also $\widehat{S}$ is a subset of the neighborhood $N_{\delta }\left(s\right),$ which in turn is a subset of the compact set ${\overline{N}}_{r/2}\left(S\right).$

Now, by part (d) of the preceding exercise, we know that Green's Theorem holds for $\widehat{S}.$ That is

We will show that Green's Theorem holds for $S$ by showing two things: (i) $|{\int }_{C_S}\omega -{\int }_{C_{\widehat{S}}}\omega |<4ϵ,$ and (ii) $|{\int }_S\left(Q_x\right)-P_y)-{\int }_{\widehat{S}}(Q_x-P_y)|<ϵ.$ We would then have, by the usual adding and subtracting business, that

and, since $ϵ$ is an arbitrary positive number, we would obtain

Let us estabish (i) first. We have from (1) above that $|u\left(x\right)-\widehat{u}\left(x\right)|<\delta$ for all $x\in [a,b],$ and from (3) that

Hence, by [link] ,

Similarly, using (2) and (4) above, we have that

Also, the difference of the line integrals of $\omega$ along the righthand boundaries of $S$ and $\widehat{S}$ is less than $ϵ.$ Thus

Of course, a similar calculation shows that

These four line integral inequalities combine to give us that

establishing (i).

Finally, to see (ii), we just compute

This establishes (ii), and the proof is complete.

At last, we can finish the proof of this remarkable result.

Proof of green's theorem

As usual, let $S$ be determined by the interval $[a,b]$ and the two bounding functions $u$ and $l.$ Recall that $u\left(x\right)-l\left(x\right)>0$ for all $x\in (a,b).$ For each natural number $n>2,$ let $S_n$ be the geometric set that is determined by the interval $[a+1/n,b-1/n]$ and the two bounding functions $u_n$ and $l_n,$ where $u_n=u-(u-l)/n$ restricted to the interval $[a+1/n,b-1/n],$ and $l_n=l+(u-l)/n$ restricted to $[a+1/n,b-1/n].$ Then each $S_n$ is a piecewise smooth geometric set, whose boundary has finite length, and each $S_n$ is contained in the open set $S^0$ where by hypothesis $\omega$ is smooth. Hence, by Lemma 4, Green's Theorem holds for each $S_n.$ Now it should follow directly, by taking limits, that Green's Theorem holds for $S.$ In fact, this is the case, and we leave the details to the exercise that follows.

REMARK Green's Theorem is primarily a theoretical result. It is rarely usedto “compute” a line integral around a curve or an integral of a functionover a geometric set. However, there is one amusing exception to this, and that iswhen the differential form $\omega =xdy.$ For that kind of $\omega ,$ Green's Theorem says that the area of the geometric set $S$ can be computed as follows:

This is certainly a different way of computing areas of sets from the methods we developed earlier. Try this way out on circles, ellipses, and the like.