Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd Edition

Chapter 4Mathematical preliminaries

1 INTRODUCTION

Chapters 4–7, which constitute Part Two of this monograph, consist of two principal parts. The first part discusses differentials and the second part deals with extremum problems.

The use of differentials in both applied and theoretical work is widespread, but a satisfactory treatment of differentials is not so widespread in textbooks on economics and mathematics for economists. Indeed, some authors still claim that dx and dy stand for ‘infinitesimally small changes in x and y’. The purpose of Chapters 5 and 6 is therefore to provide a systematic theoretical discussion of differentials.

We begin, however, by reviewing some basic concepts which will be used throughout.

2 INTERIOR POINTS AND ACCUMULATION POINTS

Let c be a point in ℝⁿ and r a positive number. The set of all points x in ℝⁿ whose distance from c is less than r is called an n‐ball of radius r and center c, and is denoted by B(c) or B(c; r). Thus,

An n‐ball B(c) is sometimes called a neighborhood of c, denoted by N(c). The two words are used interchangeably.

Let S be a subset of ℝⁿ, and assume that c ∈ S and x ∈ ℝⁿ, not necessarily in S. Then,

1. (a) if there is an n‐ball B(c), all of whose points belong to S, then c is called an interior point of S;
2. (b) if every n‐ball B(x) contains at least one point of S distinct from x, then x is called an accumulation point of S