Analysis underpins calculus, much as calculus underpins virtually all mathematical sciences. A sound understanding of analysis' results and techniques is therefore valuable for a wide range of disciplines both within mathematics itself and beyond its traditional boundaries. This text seeks to
develop such an understanding for undergraduate students on mathematics and mathematically related programmes. Keenly aware of contemporary students' diversity of motivation, background knowledge and time pressures, it consistently strives to blend beneficial aspects of the workbook, the formal
teaching text, and the informal and intuitive tutorial discussion.
The authors devote ample space and time for development of confidence in handling the fundamental ideas of the topic. They also focus on learning through doing, presenting a comprehensive range of examples and exercises, some worked through in full detail, some supported by sketch solutions and
hints, some left open to the reader's initiative.
Without undervaluing the absolute necessity of secure logical argument, they legitimise the use of informal, heuristic, even imprecise initial explorations of problems aimed at deciding how to tackle them. In this respect they authors create an atmosphere like that of an apprenticeship, in which the
trainee analyst can look over the shoulder of the experienced practitioner.
Author(s): Aisling McCluskey, Brian McMaster
Publisher: Oxford University Press
Year: 2018
Language: English
Commentary: True PDF
Pages: 400
Cover
Undergraduate Analysis: A Working Textbook
Copyright
Dedication
Preface
Contents
A Note to the Instructor
A Note to the Student Reader
1 Preliminaries
1.1 Real numbers
1.2 The basic rules of inequalities — a checklist of things you probably know already
1.3 Modulus
1.4 Floor
2 Limit of a sequence — an idea, a definition, a tool
2.1 Introduction
2.2 Sequences, and how to write them
2.3 Approximation
2.4 Infinite decimals
2.5 Approximating an area
2.6 A small slice of π
2.7 Testing limits by the definition
2.8 Combining sequences; the algebra of limits
2.9 POSTSCRIPT: to infinity
2.10 Important note on ‘elementary functions’
3 Interlude: different kinds of numbers
3.1 Sets
3.2 Intervals, max and min, sup and inf
3.3 Denseness
4 Up and down — increasing and decreasing sequences
4.1 Monotonic bounded sequences must converge
4.2 Induction: infinite returns for finite effort
4.3 Recursively defined sequences
4.4 POSTSCRIPT: The epsilontics game — the ‘fifth factor of difficulty’
5 Sampling a sequence — subsequences
5.1 Introduction
5.2 Subsequences
5.3 Bolzano-Weierstrass: the overcrowded interval
6 Special (or specially awkward) examples
6.1 Introduction
6.2 Important examples of convergence
7 Endless sums — a first look at series
7.1 Introduction
7.2 Definition and easy results
7.3 Big series, small series: comparison tests
7.4 The root test and the ratio test
8 Continuous functions — the domain thinks that the graph is unbroken
8.1 Introduction
8.2 An informal view of continuity
8.3 Continuity at a point
8.4 Continuity on a set
8.5 Key theorems on continuity
8.6 Continuity of the inverse
9 Limit of a function
9.1 Introduction
9.2 Limit of a function at a point
10 Epsilontics and functions
10.1 The epsilontic view of function limits
10.2 The epsilontic view of continuity
10.3 One-sided limits
11 Infinity and function limits
11.1 Limit of a function as x tends to infinity or minus infinity
11.2 Functions tending to infinity or minus infinity
12 Differentiation — the slope of the graph
12.1 Introduction
12.2 The derivative
12.3 Up and down, maximum and minimum: for differentiable functions
12.4 Higher derivatives
12.5 Alternative proof of the chain rule
13 The Cauchy condition — sequences whose terms pack tightly together
13.1 Cauchy equals convergent
14 More about series
14.1 Absolute convergence
14.2 The ‘robustness’ of absolutely convergent series
14.3 Power series
15 Uniform continuity — continuity’s global cousin
15.1 Introduction
15.2 Uniformly continuous functions
15.3 The bounded derivative test
16 Differentiation — mean value theorems, power series
16.1 Introduction
16.2 Cauchy and l’Hôpital
16.3 Taylor series
16.4 Differentiating a power series
17 Riemann integration — area under a graph
17.1 Introduction
17.2 Riemann integrability — how closely can rectangles approximate areas under graphs?
17.3 The integral theorems we ought to expect
17.4 The fundamental theorem of calculus
18 The elementary functions revisited
18.1 Introduction
18.2 Logarithms and exponentials
18.3 Trigonometric functions
19 Exercises: for additional practice
Suggestions for further reading
Index
