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.