6.6 Integration around closed curves, and green's theorem

Thus far, we have discussed integration over curves joining two distinct points $z_1$ and $z_2.$ Very important in analysis is the concept of integrating around a closed curve, i.e., one that starts and ends at the same point. There is nothing really new here; the formulas forall three kinds of integrals we have defined will look the same, in the sense that they all are described interms of some parameterization $\phi .$ A parameterization $\phi :[a,b]\to C$ of a closed curve $C$ is just like the parameterization for a curve joining two points, except that the two points $\phi \left(a\right)$ and $\phi \left(b\right)$ are equal.

Two problems are immediately apparent concerning integrating around a closed curve.First, where do we start on the curve, which point is the initial point? And second, which way to we go around the curve?Recall tha if $\phi :[a,b]\to C$ is a parameterization of $C,$ then $\psi :[a,b]\to C,$ defined by $\psi \left(t\right)=\phi (a+b-t),$ is a parameterization of $C$ that is the reverse of $\phi ,$ i.e., it goes around the curve in the other direction.If we are integrating with respect to arc length, this reverse direction won't make a difference, but, for contour integrals and line integrals,integrating in the reverse direction will introduce a minus sign.

The first question mentioned above is not so difficult to handle. It doesn't really matter where we start on a closed curve; the parameterization can easily be shifted.

The question of which way we proceed around a closed curve is one that leads to quite intricate and difficult mathematics, at least when we consider totaly general smooth curves.For our purposes it wil, suffice to study a special kind of closed curve, i.e., curves that are the boundaries of piecewise smooth geometric sets.Indeed, the intricate part of the general situation has a lot to do with determining which is the “inside” of the closed curve and which is the “outside,”a question that is easily settled in the case of a geometric set. Simple pictures make this general question seem silly, but precise proofs that there is a definite inside and a definite outside are difficult, and eluded mathematicians for centuries,culminating in the famous Jordan Curve Theorem, which asserts exactly what our intuition predicts: