University Mathematics Handbook (2015)
VII. Differential Calculus of Multivariable Functions
Chapter 1. Introduction
1.1 Domain in
a. is a point on . are its coordinates.
b. The distance between and
is .
c. An -dimensional sphere of radius , with its center at is the locus of all points the distance of which from is smaller than or equal to , and which hold: .
d. A sphere is an open sphere if this is a strict inequality.
e. neighborhood of is an open sphere with radius centered at .
f. is an interior point of set of if there exists -neighborhood of which is entirely at .
g. is an open set in if all is points are interior points.
h. Set is bounded if there exists a sphere of a finite radius containing it.
i. Continuous line at is the locus of all points of coordinates when are continuous functions of parameter .
j. is a connected set if every two interior points of it can be connected by a continuous line which is entirely at .
k. An open and connected set is called a domain.
1.2 Sequences of Points
a. Limit Point of Sequence
is point , if for every there exists such that for all there holds . It is denoted .
b. Set of points converges to if and only if sequences of coordinates converge to coordinates , respectively.
c. Bolzano-Weierstrass Theorem: every bounded and infinite sequence of points at has a subset converging to the limit.