Connectivity of regions

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 = a2 (1 − sin 2φ), the line integral becomes
π
I = (x − y)2 ds =
a3 (1 − sin 2φ) dφ = πa3 . C
0
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
u1 , u2 , u3 is
(ds)2 = h21 (du1 )2 + h22 (du2 )2 + h23 (du3 )2 ,
where h1 , h2 , h3 are the scale factors of the coordinate system. If a curve C in
three dimensions is given parametrically by the equations ui = ui (λ) for i = 1, 2, 3
then the element of arc length along the curve is
du1 2
du2 2
du3 2
+ h22
+ h23
dλ.
ds = h21
dλ
dλ
dλ
11.2 Connectivity of regions
In physical systems it is usual to define a scalar or vector field in some region R.
In the next and some later sections we will need the concept of the connectivity
of such a region in both two and three dimensions.
We begin by discussing planar regions. A plane region R is said to be simply
connected if every simple closed curve within R can be continuously shrunk to
a point without leaving the region (see figure 11.2(a)). If, however, the region
R contains a hole then there exist simple closed curves that cannot be shrunk
to a point without leaving R (see figure 11.2(b)). Such a region is said to be
doubly connected, since its boundary has two distinct parts. Similarly, a region
with n − 1 holes is said to be n-fold connected, or multiply connected (the region
in figure 11.2(c) is triply connected).
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