Page 126 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 6] PARTIAL DERIVATIVES 117

NEIGHBORHOODS
The set of all points ðx; yÞ such that jx x 0 j < , j y y 0 j < where > 0, is called a rectangular
neighborhood of ðx 0 ; y 0 Þ; the set 0 < jx x 0 j < ,0 < j y y 0 j < which excludes ðx 0 ; y 0 Þ is called a
rectangular deleted neighborhood of ðx 0 ; y 0 Þ. Similar remarks can be made for other neighborhoods,
2
2
2
e.g., ðx x 0 Þ þð y y 0 Þ < is a circular neighborhood of ðx 0 ; y 0 Þ. The term ‘‘open ball’’ is used to
designate this circular neighborhood. This terminology is appropriate for generalization to more
dimensions. Whether neighborhoods are viewed as circular or square is immaterial, since the descrip-
tions are interchangeable. Simply notice that given an open ball (circular neighborhood) of radius
p ffiffiffi
there is a centered square whose side is of length less than 2 that is interior to the open ball, and
conversely for a square of side there is an interior centered of radius of radius less than =2. (See Fig.
6-1.)
A point ðx 0 ; y 0 Þ is called a limit point, accumulation point, or cluster point of a point set S if every
deleted neighborhood of ðx 0 ; y 0 Þ contains points of S.Asin the case of one-dimensional point sets,
every bounded infinite set has at least one limit point (the Bolzano–Weierstrass theorem, see Pages 6 and
12). A set containing all its limit points is called a closed set.

Fig. 6-1 Fig. 6-2

REGIONS
A point P belonging to a point set S is called an interior point of S if there exists a deleted
neighborhood of P all of whose points belong to S.A point P not belonging to S is called an exterior
point of S if there exists a deleted neighborhood of P all of whose points do not belong to S.A point P
is called a boundary point of S if every deleted neighborhood of P contains points belonging to S and
also points not belonging to S.
If any two points of a set S can be joined by a path consisting of a finite number of broken line
segments all of whose points belong to S, then S is called a connected set.A region is a connected set
which consists of interior points or interior and boundary points. A closed region is a region containing
all its boundary points. An open region consists only of interior points. The complement of a set, S,in
the x y plane is the set of all points in the plane not belonging to S. (See Fig. 6-2.)
Examples of some regions are shown graphically in Figs 6-3(a), (b), and (c) below. The rectangular
region of Fig. 6-1(a), including the boundary, represents the sets of points a @ x @ b, c @ y @ d which
is a natural extension of the closed interval a @ x @ b for one dimension. The set a < x < b, c < y < d
corresponds to the boundary being excluded.
In the regions of Figs 6-3(a) and 6-3(b), any simple closed curve (one which does not intersect itself
anywhere) lying inside the region can be shrunk to a point which also lies in the region. Such regions are
called simply-connected regions.In Fig. 6-3(c) however, a simple closed curve ABCD surrounding one of
the ‘‘holes’’ in the region cannot be shrunk to a point without leaving the region. Such regions are called
multiply-connected regions.