id: 06518675
dt: b
an: 2016e.00802
au: Geveci, Tunc
ti: Advanced calculus of a single variable.
so: Cham: Springer (ISBN 978-3-319-27806-3/hbk; 978-3-319-27807-0/ebook). xii,
382~p. (2016).
py: 2016
pu: Cham: Springer
la: EN
cc: I15
ut: Cauchy sequence; continuous function, intermediate value property;
integrability
ci:
li: doi:10.1007/978-3-319-27807-0
ab: This volume is devoted to a thorough discussion of some basic concepts and
theorems related to a beginning calculus course. The author is mainly
interested to the development of the fundamental concepts of calculus,
such as limits (with an emphasis on $ε$-$δ$ definitions), continuity
(including an appreciation of the difference between mere pointwise and
uniform continuity), the derivative (with rigorous proofs of various
versions of L’Hôpital’s rule) and the Riemann integral (discussing
improper integrals, comparison and Dirichlet tests). The content of the
book under review is divided into the following six chapters: 1. Real
numbers, sequences, and limits; 2. Limits and continuity of functions;
3. The derivative; 4. The Riemann integral; 5. Infinite series; 6.
Sequences and series of functions. The first chapter begins with a
quick review of the properties of the set of real numbers as an ordered
field. The concept of the limit of a sequence and the relevant rules
are discussed rigorously. The completeness of the field of real numbers
is introduced as the existence of the limit of a Cauchy sequence. The
least upper bound principle and the special nature of the convergence
or divergence of a monotone sequence are also treated in Chapter 1.
Even the notion of an infinite limit is also discussed here. In the
next chapter, the author is interested in the development of the basic
notions of continuity and limits of functions. The discussion is
limited to functions defined on intervals. A central role in this
development is played by the $ε$-$δ$ definitions, which go back to
the pioneering work of Cauchy. One of the highlights of the chapter is
the intermediate value property, which has bearing on the definitions
of basic inverse functions that figure prominently in beginning
calculus. Chapter 3 is devoted to the notion of derivative and some of
its main properties. Furthermore, convexity as a nice application of
the mean value theorem is discussed in detail. Rigorous proofs of
various versions of the L’Hôpital rule are also included in this
chapter. Chapter 4 is concerned with the Riemann integral.
Integrability criteria in terms of upper and lower sums and the
oscillation of a function are discussed. A detailed discussion of
improper integrals, including comparison and Dirichlet tests, is also
included. Chapter 5 is a review of series of real numbers. A nice part
in this chapter includes Cauchy-type criteria for the convergence of
series. Chapter 6 discusses the convergence of sequences and series of
real-valued functions on intervals. The distinction between mere
pointwise and uniform convergence is emphasized with ample examples.
The analyticity of functions defined via power series follows smoothly
once the appropriate foundation involving uniform convergence is
established. The chapter concludes with the definition of familiar
special functions via power series. The presentation is thorough and
clear with many comments on the historical context of the problems and
concepts. Requiring only basic knowledge of elementary calculus, this
book presents the necessary material for students and professionals in
various mathematics-related fields, such as engineering, statistics,
and computer science, to explore real analysis.
rv: Teodora-Liliana Rădulescu (Craiova)