6.6 Integration around closed curves, and green's theorem  (Page 2/5)

Jordan curve theorem

The complement of a closed curve is the union of two disjoint components, one bounded and one unbounded.

We define the bounded component to be the inside of the curve and the unbounded component to be the outside.

We adopt the following convention for how we integrate around the boundary of a piecewise smooth geometric set $S.$ That is, the curve $C_S$ will consist of four parts: the lower boundary (graph of the lower bounding function $l$ ), the righthand boundary (a portion of the vertical line $x=b$ ), the upper boundary (the graph of the upper bounding function $u$ ), and finally the lefthand side (a portion of the vertical line $x=a$ ). By integrating around such a curve $C_S,$ we will always mean proceeding counterclockwise around the curves. Specifically, we move from left to right along the lower boundary, from bottom to top alongthe righthand boundary, from right to left across the upper boundary, and from top to bottom along the lefthand boundary.Of course, as shown in the exercise above, it doesn't matter where we start.

The next theorem, though simple to state and use, contains in its proof a combinatorial idea that is truly central to all that follows in this chapter.In its simplest form, it is just the realization that the line integral in one direction along a curveis the negative of the line integral in the opposite direction.

Let $S_1,...,S_n$ be a collection of closed geometric sets that constitute a partition of a geometric set $S,$ and assume that the boundaries of all the $S_i$ 's, as well as the boundary of $S,$ have finite length. Suppose $\omega$ is a continuous differential form on all the boundaries $\left\{C_{S_k}\right\}.$ Then

We give a careful proof for a special case, and then outline the general argument. Suppose then that $S$ is a piecewise smooth geometric set, determined by the interval $[a,b]$ and the two bounding functions $u$ and $l,$ and assume that the boundary $C_S$ has finite length. Suppose $m\left(x\right)$ is a piecewise smooth function on $[a,b],$ satisfying ${\int }_a^b\left|m^{\text{'}}\right|<\infty ,$ and assume that $l\left(x\right)<m\left(x\right)<u\left(x\right)$ for all $x\in (a,b).$ Let $S_1$ be the geometric set determined by the interval $[a,b]$ and the two bounding functions $m$ and $l,$ and let $S_2$ be the geometric set determined by the interval $[a,b]$ and the two bounding functions $u$ and $m.$ We note first that the two geometric sets $S_1$ and $S_2$ comprise a partition of the geometric set $S,$ so that this is indeed a pspecial case of the theorem.

Next, consider the following eight line integrals: First, integrate from left to write along the graph of $m,$ second, up the line $x=b$ from $(b,m(b\left)\right)$ to $(b,u(b\left)\right),$ third, integrate from right to left across the graph of $u,$ fourth, integrate down the line $x=a$ from $(a,u(a\left)\right)$ to $(a,m(a\left)\right),$ fifth, continue down the line $x=a$ from $(a,m(a\left)\right)$ to $(a,l(a\left)\right),$ sixth, integrate from left to right across the graph of $l,$ seventh, integrate up the line $x=b$ from $(b,l(b\left)\right)$ to $(b,m(b\left)\right),$ and finally, integfrate from right to left across the graph of $m.$