Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.2 CONNECTIVITY OF REGIONS

(a) (b) (c)

Figure 11.2 (a) A simply connected region; (b) a doubly connected region;

(c) a triply connected region.

Since (x−y)^2 =a^2 (1−sin 2φ), the line integral becomes

I=

∫

C

(x−y)^2 ds=

∫π

0

a^3 (1−sin 2φ)dφ=πa^3 .

As discussed in the previous chapter, the expression (10.58) for the square of

the element of arc length in three-dimensional orthogonal curvilinear coordinates

u 1 ,u 2 ,u 3 is

(ds)^2 =h^21 (du 1 )^2 +h^22 (du 2 )^2 +h^23 (du 3 )^2 ,

whereh 1 ,h 2 ,h 3 are the scale factors of the coordinate system. If a curveCin

three dimensions is given parametrically by the equationsui=ui(λ)fori=1, 2 , 3

then the element of arc length along the curve is

ds=

√

h^21

(

du 1

dλ

) 2

+h^22

(

du 2

dλ

) 2

+h^23

(

du 3

dλ

) 2

dλ.

11.2 Connectivity of regions

In physical systems it is usual to define a scalar or vector field in some regionR.

In the next and some later sections we will need the concept of theconnectivity

of such a region in both two and three dimensions.

We begin by discussing planar regions. A plane regionRis said to besimply

connectedif every simple closed curve withinRcan be continuously shrunk to

a point without leaving the region (see figure 11.2(a)). If, however, the region

Rcontains a hole then there exist simple closed curves that cannot be shrunk

to a point without leavingR(see figure 11.2(b)). Such a region is said to be

doubly connected, since its boundary has two distinct parts. Similarly, a region

withn−1 holes is said to ben-fold connected,ormultiply connected(the region

in figure 11.2(c) is triply connected).