Introduction to Analysis, 2/e
William R. Wade, University of Tennessee
Published July, 1999 by Prentice Hall Engineering/Science/Mathematics
Copyright 2000, 611 pp.
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For one/two-semester, junior/senior-level courses in Advanced
Calculus, Analysis I, Real Analysis taken by math majors. The first
semester is usually a general requirement; the second semester is
often an option for the more motivated students.
Designed to challenge advanced students while bringing weaker
students up to speed, this text is an advantageous alternative to
most other analysis texts which either tend to be too easy
(designed for an Intermediate Analysis course) or too difficult
(designed for students headed for a Ph.D. in Pure Mathematics)both
of which usually tend to slight multidimensional material. Hailed
for its readability, practicality, and flexibility, this text presents
the Fundamental Theorems from a very practical point of view. Introduction
to Analysis starts slowly and carefully, with a focused presentation
of the material; saves extreme abstraction for the second semester;
provides optional enrichment sections; includes many routine exercises
and examples; and liberally supports (with examples and hints) what
little theory is developed in the exercises.
NEWSeparate coverage of topology and analysisPresents
purely computational material first, in its own chapter (Ch. 8); then
develops the topological material two alternate chapters (Ch. 9 on
Euclidean spaces, and Ch. 9 on Metric spaces).
NEWTightened level of rigor in the presentation
of integers (Ch. 1)Relegates all well known properties
to the Appendices.
- Provides a much better flow, makes topology easier to
NEWRevised coverage on limitsMakes
coverage of the concepts of limits supremum and infimum optional.
Eliminates limits through sets and the concept of cluster
point (except where it is essential in the metric space chapter).
Relegates the concept sequentially compact to the exercises,
where it has also been made optional.
- Provides a shorter presentation and focuses the text
even more on analysis.
NEWReorganized coverage of seriesSeparates
series of constants and series of functions into two separate chapters.
Includes many more routine exercises, and eliminates some of the more
bizarre problems. Segregates several of the more esoteric tests (e.g.,
the log test, Raabe's test) into optional sections.
NEWMany more elementary exercisesIn
the chapters on differentiability and integrability, especially in
the multidimensional sections.
NEWConsecutive numbering of Theorems,
Definitions, Remarks, etc.
A uniform writing style and notationThroughout.
Presents material in small chunks (Remarks and Examples), with
larger sections presenting an integrated point of view.
- Allows students and instructors to more easily find citations.
Unusually friendly, but rigorous treatmentStarts
slowly (to give less capable students time to get up to speed),
but contains all the major results associated with the standard material
called Advanced Calculus. Uses analogy and geometry
to motivate and explain the theory. Precedes many complicated proofs
with a paragraph labeled Strategy which motivates
the proof, shows why it was chosen, and why it should work. Follows
many theorems with examples to show why each hypothesis is needed,
allowing students to remember the hypotheses by recalling the examples.
Presents proofs in complete detail, with each step carefully
documented. Links proofs together in a way that teaches students
to think ahead: Why This Theorem? Uses physical
interpretations to examine some concepts from a second or third
point of view.
Early introduction of the fundamental goals of analysisAnd
references to them oftene.g., interchanging two infinite operations
(e.g., a limit sign and an integral sign), or examining how a limit
operation interacts with an algebraic operation (e.g., set union or
- Meets students where they are, giving the weaker
students all the support they need to develop basic requisite skills
and knowledge, but not getting in the way of those who want
a quicker pace and deeper focus.
A practical focusThe text is honest and explicit
about what is assumed from the very beginning.
- Students know what can be used, what proved.
More coverage of multidimensional analysis than in most
other textsProvides an honest, geometrically motivated treatment
of the calculus of several variables, which includes the Fundamental
Theorems of Vector Calculusa topic crucial to applied mathematicians
that is usually glossed over in other texts or buried in an elegant
algebraic presentation that is overwhelming to average undergraduates.
- Enables students to see that many separate theorems are
part of a single point of view.
An alternate chapter on metric spacesDesigned
for instructors who feel these concepts are best presented first in
generality, instead of in the concrete Euclidean space setting.
- Prepare students to study subfields like harmonic analysis
and partial differential equations, where knowledge of vector calculus
(especially, how actually to compute line and surface integrals instead
of just throwing these terms around) is needed. These topics are in
the forefront of contemporary research and are essential for students
heading to graduate school in any quantitative field of study.
Optional appendices and Enrichment sections.
- Enables serious students to delve further into
the material and allows instructors to tailor their courses
to their own tastes and student needs. Also keeps the main text from
getting bogged down in esoterica (e.g., tests for convergence, partitions
of unity, etc.).
Over 200 worked examples.
- Allows instructors to cover either chapter independentlywithout
mentioning the other. All references to material from the topological
chapter refers to BOTH these chapters, so students can skip
either one of those chapters without loss of integrity.
Over 600 exercisesRanging from simple to challenging,
including many applied-type exercises. Some are computational, some
theoretical, some interpret the theory in a classical or practical
way, some develop peripheral theory. Special boxed exercises develop
theory that is used later in the book. Difficult exercises are presented
in several stepswhich serve as an outline to a solution.
- Helps students review, shows them several different ways
to apply a given result, or shows them the limitations of the theory
(e.g., what happens to a result when one of the hypotheses is missing).
- Allows students to test their comprehension of concepts,
to focus clearly on the most important concepts/techniques, and to
use the material in other contexts.
I. ONE-DIMENSIONAL THEORY.
1. The Real Number System.
2. Sequences in R.
3. Continuity on R.
4. Differentiability on R.
5. Integrability on R.
6. Infinite Series of Real Numbers.
7. Infinite Series of Functions.
II. MULTIDIMENSIONAL THEORY.
8. Euclidean Spaces.
9. Topology of Euclidean Spaces.
10. Metric Spaces.
11. Differentiability on Rn.
12. Integration on Rn.
13. Fundamental Theorems of Vector Calculus.
14. Fourier Series.
15. Differentiable Manifolds.
Answers and Hints to Exercises.