Page 241 - Schaum's Outline of Theory and Problems of Advanced Calculus

232 LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS [CHAP. 10

SIMPLE CLOSED CURVES, SIMPLY AND MULTIPLY CONNECTED REGIONS
A simple closed curve is a closed curve which does not intersect itself anywhere. Mathematically, a
curve in the xy plane is defined by the parametric equations x ¼ ðtÞ; y ¼ ðtÞ where and are single-
valued and continuous in an interval t 1 @ t @ t 2 .If ðt 1 Þ¼ ðt 2 Þ and ðt 1 Þ¼ ðt 2 Þ, the curve is said to
be closed.If ðuÞ¼ ðvÞ and ðuÞ¼ ðvÞ only when u ¼ v (except in the special case where u ¼ t 1 and
v ¼ t 2 ), the curve is closed and does not intersect itself and so is a simple closed curve. We shall also
assume, unless otherwise stated, that and are piecewise differentiable in t 1 @ t @ t 2 .
If a plane region has the property that any closed
curve in it can be continuously shrunk to a point
without leaving the region, then the region is called
simply connected; otherwise, it is called multiply con-
nected (see Fig. 10-2 and Page 118 of Chapter 6).
As the parameter t varies from t 1 to t 2 , the plane
curve is described in a certain sense or direction.
Fig. 10-2
For curves in the xy plane, we arbitrarily describe
this direction as positive or negative according as a person traversing the curve in this direction with his
head pointing in the positive z direction has the region enclosed by the curve always toward his left or
right, respectively. If we look down upon a simple closed curve in the xy plane, this amounts to saying
that traversal of the curve in the counterclockwise direction is taken as positive while traversal in the
clockwise direction is taken as negative.

GREEN’S THEOREM IN THE PLANE
This theorem is needed to prove Stokes’ theorem (Page 237). Then it becomes a special case of that
theorem.
Let P, Q, @P=@y;@Q=@x be single-valued and continuous in a simply connected region r bounded by
a simple closed curve C. Then
ðð
@Q @P
þ
dx dy
Pdx þ Qdy ¼ ð10Þ
C @x @y
r
þ
where is used to emphasize that C is closed and that it is described in the positive direction.
C
This theorem is also true for regions bounded by two or more closed curves (i.e., multiply connected
regions). See Problem 10.10.
CONDITIONS FOR A LINE INTEGRAL TO BE INDEPENDENT OF THE PATH
The line integral of a vector field A is independent of path if its value is the same regardless of the
(allowable) path from initial to terminal point. (Thus, the integral is evaluated from knowledge of the
coordinates of these two points.)
For example, the integral of the vector field A ¼ yi þ xj is independent of path since
ð ð ð
x 2 y 2
dðxyÞ¼ x 2 y 2 x 1 y 1
A dr ¼ ydx þ xdy ¼
C C x 1 y 1
Thus, the value of the integral is obtained without reference to the curve joining P 1 and P 2 .
This notion of the independence of path of line integrals of certain vector fields, important to theory
and application, is characterized by the following three theorems:
ð
Theorem 1. A necessary and sufficient condition that A dr be independent of path is that there
exists a scalar function such that A ¼r . C