Hi, wondering-ga:
I'm going ahead to post this answer with some of the intended
editorial text to wrap the last two sets of selections not yet
finished, because the question is going to expire soon. I'll supply
those remarks on Combinatorics and Measure/integration theory
tomorrow.
I've taken the liberty of rearranging the order of subjects in a
manner that to some extent reflects the relative "mathematical
maturity" expected of students and which also always an opportunity
for me to discuss the overlaps in subject matter efficiently.
regards, mathtalk-ga
Introduction
============
Before diving into the book lists themselves, let me present three
general on-line references to math books and subject matter that I
consulted extensively in preparing the answer (in addition to use of
my own experiences as student and instructor).
The first is a "mathematical atlas" site maintained by Prof. Dave
Rusin of Northern Illinois University. I've incoroporated a number of
additional links to his site in what follows. For some areas he
recommends for books for undergraduate courses. Where he fails to do
so, it highlights to some degree the difficulty of drawing a boundary
between "advanced" undergraduate and first-year graduate course
material. It is nevertheless helpful in keeping the relationships and
boundaries between fields of mathematics in perspective.
[Dave Rusin's Known Math: Tour of the subfields of mathematics]
http://www.math.niu.edu/~rusin/known-math/index/tour.html
Eric Weisstein is well known for his on-line encyclopedia of
mathematics. Here's a link to a less known resource of books
categorized by specific math topics. Unfortunately there's no
specific treatment of undergraduate textbooks.
[Eric Weisstein's Encyclopedia of Scientific Books (Math)]
http://www.ericweisstein.com/encyclopedias/books/Mathematics.html
This is one publisher, well known for their editorial devotion to
mathematics, who has an "undergraduate" series for many subjects.
However a better characterization of the style in this series might be
that they're directed to non-specialists interested in a topic, rather
than being aimed squarely at undergraduate course adoption.
[Springer-Verlag: Undergraduate Series in Mathematics]
http://www.springeronline.com/sgw/cda/frontpage/0,10735,5-10042-69-1186407-0,00.html
Calculus = Calculus of the infinitesimal
----------------------------------------
The calculus of the infinitesimal represents both an introduction to
and a summit of difficulty in the analytic areas of mathematics. One
here confronts the ancient paradoxes of motion and infinity, and a
practical if not clearly revealed resolution is obtained.
[Thomas' Calculus (10th Ed.) by George B. Thomas et al]
http://textkit.com/support-textkit/support-item_id-0201755270-search_type-AsinSearch-locale-us.html
My top pick is the current edition of a "classic" textbook by MIT's
George Thomas. The book's tradition has been carried on mainly by R.
S. Finney, though recently additional co-authors have been added. I
own copies of the 3rd and 5th editions, and by placing it first I'm
making a sort of reactionary protest against a "dumbing down" trend in
undergraduate texts that began in the late 70's/early 80's.
The second and third texts listed are epitomes of what the publishing
houses successfully market to the college audiences, the "big thick
books" of calculus. Anton, as we will discuss in a moment, rose to
prominence through the wide acceptance of his book in linear algebra.
"Early" transcendentals refers to a teaching schedule that presents
the elementary transcendental functions (trig, log/exp) ahead of where
they would normally have been given in a traditional arrangement (for
math/engineering students), so this would actually be the variation
used in some calculus for business students classes.
[Calculus: Early Transcendentals (7th Ed.) by Howard A. Anton et al]
http://jws-edcv.wiley.com/college/bcs/redesign/student/0,,_0471445967_BKS_1344____,00.html
[Calculus with Analytic Geometry (6th Ed.) by Edwards and Penney]
http://www.rbookshop.com/mathematics/c/Calculus/Calculus_with_Analytic_Geometry_0130920711.htm
Any of these three I think would fulfill your criteria for an
undergraduate college calculus text with significant
popularity/acceptance. For a book often used in an honors calculus
setting not long ago, see:
[Calculus by Michael Spivak]
http://www.mathpop.com/bookhtms/cal.htm
Manifold-ga notes that to a great extent calculus has become a
high-school subject, as is evidenced by the widespread use of the AP
calculus exams, But I think a distinction can still be made between
the comprehensiveness expected at the two levels, especially for a 2nd
or 3rd term of college calculus text.
This link to Dave Rusin's Known Math (aka The Mathematical Atlas) is a
page describing one of four main subfields of mathematics, namely
analyis. You can drill down from there to the Calculus specific
subpage, and it serves to give an overview of what I think of as one
"half" of mathematics.
[Dave Rusin's Known Math: Analytic areas of mathematics]
http://www.math.niu.edu/~rusin/known-math/index/tour_ana.html
Algebra = Abstract algebra (or "modern" algebra)
-------------------------------------------------
Just as Calculus is totemic of analysis and the "continuous" side of
mathematics, so too does the undergraduate course in "abstract" or
"modern" algebra introduce another side, symbolic and discrete in
nature. Algebra closely intertwines with analysis in many places, and
I refer to it as the second "half" of mathematics, reserving the
opportunity for a joke later on about the "third half".
The undergraduate abstract algebra texts I first studied (one by
Paley, another by McCoy) are out of print, and in college my abstract
algebra instructor taught from his mimeographed notes. I'm making my
top recommendation based on having read some on-line chapters from the
book during its preparation.
[Abstract Algebra (2nd Ed.) by Beachy and Blair]
http://www.math.niu.edu/~beachy/abstract_algebra/frames_index.html
Serge Lang is a good writer, in my opinion, known for being outspoken
and challenging orthodoxy on frequent occasions.
[Undergraduate Algebra by Serge A. Lang]
http://www.bookhq.com/compare/038797279X.html
My third choice aims at an approach that presents abstract algebra as
less of a "pure" math topic and more of an "applied" one. With the
larger role for computer algorithms of a discrete nature, e.g.
cryptography, this has already become something of a quiet revolution
in mathematical education, i.e. a focus on computer-aided algebra.
[Modern Algebra with Applications (2nd Ed.) by Gilbert and Nicholson]
http://www.wileyeurope.com/WileyCDA/WileyTitle/productCd-0471414514.html
A fourth choice is one mentioned by manifold-ga and might fall in the
"honors" course category. The author died in 1988, and the 3rd
edition was a revision in 1996 by David J. Winter. Herstein's
treatment of elementary group theory is considered especially
polished.
[Abstract Algebra (3rd Ed.) by I. N. Herstein]
http://www.amazon.com/exec/obidos/tg/detail/-/0471368792?v=glance
Again I've provided a link to a rather broad subject matter page in
"The Mathematical Atlas", rather than one to a more specific "abstract
algebra" page. In fact Dave doesn't really match this category
precisely, pointing out that his two pages on Groups and on Rings and
Fields are most clearly within that realm, and many other related
areas crowd into the picture. An introductory course in abstract
algebra, for example, will often serve as an introduction to
combinatorial aspects like "the pigeonhole principle", to number
theory's "mathematical induction", and to "naive" set theory in which
much of mathematics seems most naturally founded.
[Dave Rusin's Known Math: Algebraic areas of mathematics]
http://www.math.niu.edu/~rusin/known-math/index/tour_alg.html
Differential equations
----------------------
We now come to to course areas that I didn't actually take as an
undergraduate, but which I've taught numerous times. My top choice, a
textbook I taught from, seems to have wide acceptance at the second
year level:
[Elementary Differential Equations and BVP (7e) by Boyce and DiPrima]
http://search.barnesandnoble.com/textbooks/booksearch/isbninquiry.asp?userid=36MLS1VRT9&pwb=1&ean=9780471319993
Given the consenus on that text, I provide two additional choices
mainly to illustrate how some of this material can be classified. One
big distinction is between linear and nonlinear equations. Another is
between ordinary and partial differential equations. Also one may
contrast single differential equations with systems of them (in
multiple unknown functions). Often one approximates a solution to a
partial differential equation through a system of ordinary
differential equations, for example.
[Linear Ordinary Differential Equations by Coddington and Carlson]
http://www.addall.com/detail/0898713889.html
[Applied Partial Differential Equations by Logan and Logan]
http://www.target.com/gp/detail.html/601-5330696-2134562?asin=0387984399
The reading list suggested by benny1979-ga has a fairly
applied/analytical tilt to it, and although Boyce and DiPrima and
listed first under "302 Differential Equations" there, other texts are
also mentioned, e.g. under "303 Dynamical Systems".
See also the text suggestions under here, which include an "advanced
undergraduate text" by Robert E. O'Malley, Jr., as well as some online
manuscripts.
[Dave Rusin's Known Math: Ordinary Differential Equations]
http://www.math.niu.edu/~rusin/known-math/index/34-XX.html
Linear algebra
--------------
Here again is a subject that I didn't specifically "take" as an
undergraduate but subsequently taught quite a bit. There is the
"abstract" side of the subject (vector spaces) and the "applied" side
(matrices), and I think the relationship was made clear to me through
the various abstract algebra classes I took combined with applications
in analytic geometry and multivariable calculus.
But this has come to be a course in its own right since then, partly
due to the very successful text I taught from by Howard Anton. He did
a good job of setting the matrix manipulations (elementary row
operations, Gauss elimination) into a context for answering "abstract"
questions (linear independence, spanning) that can be defined for
general vector spaces.
[Elementary Linear Algebra (8th Ed.) by Howard A. Anton]
http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471170550.html
Now "Anton" is in some sense a commodity, being specialized into
various course "niches". For example, consider these variations on
the "original:
[Elementary Linear Algebra (8th/9th Ed.) by Anton and Rorres]
http://www.mathbook.com/a/Linear_Algebra/Elementary_Linear_Algebra__0471170526.htm
[Contemporary Linear Algebra (1st Ed.) by Anton and Busby)]
http://www.wileyeurope.com/WileyCDA/WileyTitle/productCd-0471163627.html
For a different presentation of the linear algebra material, tied into
the important application of linear differential equations (principle
of superposition), see this text:
[Linear Algebra: A First Course with App to Diff Eqn by Tom Apostol]
http://www.wileyeurope.com/WileyCDA/WileyTitle/productCd-0471174211.html
Rusin's page here hits a somewhat wider target than covered by the
books above, but it helps to place them in perspective of the more
advanced developments of this theory:
[Dave Rusin's Known Math: Linear and multilinear algebra; matrix theory]
http://www.math.niu.edu/~rusin/known-math/index/15-XX.html
Analysis = Real analysis
------------------------
For the real analysis subject I'll say that there are three well-known
"standard" undergraduate textbooks in this area, well-differentiated
from their counterpart first-year graduate texts. Of these the third
is widely considered the "hardest" in respect of Rudin's terse but
elegant style.
[Mathematical Analysis (2nd Ed.) by Tom M. Apostol]
http://www.amazon.com/exec/obidos/tg/detail/-/0201002884?v=glance
[Real Analysis (3rd Ed.) by Halsey Royden]
http://www.amazon.com/exec/obidos/tg/detail/-/0024041513?v=glance
[Principles of Mathematical Analysis (3rd Ed.) by Walter Rudin]
http://btobsearch.barnesandnoble.com/textbooks/booksearch/isbnInquiry.asp?btob=Y&isbn=007054235X
While none of these is mentioned at this page of Dave Rusin's
Mathematical Atlas, he does identify some "primers" on this topic
(including one in the Springer-Verlag undergraduate series in
mathematics), and some on-line tutorials:
[Dave Rusin's Known Math: Real functions]
http://www.math.niu.edu/~rusin/known-math/index/26-XX.html
Complex function theory = Complex analysis
------------------------------------------
An aging but acknowledged undergraduate classic (going back to the
1950's) in this area is:
[Complex Analysis by L. V. Ahlfors]
http://www.amazon.com/exec/obidos/tg/detail/-/0070006571?v=glance
Also once a very popular textbook for undergraduates in complex
analysis, now out of print:
[Complex Variables and Applications by R. Churchill and J. Brown]
http://www.target.com/gp/detail.html?asin=0070108730
Finally a fairly recent attempt to make a new presentation of this
quite important and satisfying theory:
[Introduction to Complex Analysis (2nd Ed.) by H. A. Priestly]
http://www.oup.co.uk/isbn/0-19-852562-1
Again we find no clear designation of an undergraduate text at Dave's
site, although a couple of the works he mentions are of the "primer"
category:
[Dave Rusin's Known Math: Functions of a complex variable]
http://www.math.niu.edu/~rusin/known-math/index/30-XX.html
Geometry(?) = Analytical geometry(?) Projective geometry(?)
----------- Euclidean geometry(?) Algebraic geometry(?)
Okay, geometry can mean _so_ many different things at the college
level, I'm not quite sure what tack to take. The analytical geometry
material will have been adequately covered under Calculus, so I won't
try to recover that ground. Probably the most cogent interpretation
is the kind of class offered esp. in mathematical education curricula,
where the aim is to update the high school treatment of axiomatic
Euclidean geometry with topics from a non-Euclidean perspective, such
as projective geometry. So here's a book that fills that role:
[Introduction to Geometry (2nd Ed.) by H. S. M. Coxeter]
http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471504580.html
Although Coxeter's book, which is by now an acknowledged "classic"
written in the 1960's, was reprinted fairly recently, it is too terse
in places for some tastes. A more recent book aimed at the same
undergraduate treatment of classical geometry as understood through
Klein's Erlangen program (properties invariant under transformations)
is this:
[Geometry by Brannan, Esplen, and Gray]
http://books.cambridge.org/0521597870.htm
If it were considered desirable to trim out much rehashing of the
Euclidean case already treated axiomatically in high school and
synthetically in analytic geometry, then this might be a better
choice:
[Affine and Projective Geometry by M. K. Bennett]
http://www.amazon.com/exec/obidos/tg/detail/-/0471113158?v=glance
Perhaps the most interesting link in this page of Dave's atlas:
[Dave Rusin's Known Math: Geometry]
http://www.math.niu.edu/~rusin/known-math/index/51-XX.html
is not to a textbook but to an instructional geometry software package:
[Cabri geometry]
http://www.cabri.net/cabri/index-e.html
Topology
--------
The undergraduate introduction to point set topology must necessarily
overlap with the preliminary discussions needed in both real and
complex analysis, as the inherently metric definitions of limits and
continuity are generalized to some extent to handle uniform
convergence, convergence of nets (for Riemann integration), etc.
After some searching I've settled on this as a top recommendation:
[Topological Spaces: From Distance to Neighborhood by Buskes and van Rooij]
http://www.amazon.com/exec/obidos/tg/detail/-/0387949941?v=glance
Part of the difficulty is trying to distinguish the upper division
undergraduate topology texts from ones that are used in first-year
graduate courses. Typical of the latter is:
[Topology (2nd Ed.) by James Munkres]
http://www.amazon.com/exec/obidos/tg/detail/-/0131816292?v=glance
In the "aging classic in the field" category, good as an elementary
reference and available as an inexpensive Dover paperback:
[Topology by Hocking and Young]
http://www.mathbook.com/t/Topology/Topology_0486656764.htm
[Dave Rusin's Known Math: General topology]
http://www.math.niu.edu/~rusin/known-math/index/54-XX.html
Mathematical logic
------------------
After searching a good bit I decided to make my top recommendation
this inexpensive Dover edition of the textbook I used as an
undergraduate (authored by my undergraduate instructor):
[First-order mathematical logic by Angelo Margaris]
http://shop.store.yahoo.com/doverpublications/0486662691.html
I weighed the risk that the author, being especially familiar with the
material and text, may have done an outstanding job of its
presentation.
But the classics in this field tend to be at the graduate level, eg.
Shoenfield and Kleene, but this treatment by a giant of 20th century
logic is surprisingly appropriate for undergraduates (though less
complete than Margaris's text for mathematical logic):
[Introduction to logic by Alfred Tarski]
http://www.amazon.com/exec/obidos/tg/detail/-/048628462X?v=glance
Here's one that Rusin says is "possibly" an undergraduate text:
[A mathematical introduction to logic by Herbert B. Enderton]
http://www.amazon.com/exec/obidos/tg/detail/-/0122384520?v=glance
[Dave Rusin's Known Math: Mathematical logic and foundations]
http://www.math.niu.edu/~rusin/known-math/index/03-XX.html
Probability theory (but not statistics?)
----------------------------------------
To a degree this category would have been easier to answer if
probability were lumped with statistics, for then the subject matter
would be more aligned to the needs of a service course (covering
statistical techniques) rather than the relatively pure domain of
probability theory. If statistics were the focus, I'd have cited a
standard text by Hogg and Craig (Introduction to Mathematical
Statistics) or Hogg and Tanis (Probability and Statistical Inference).
But here's a book by a former colleague of mine (friendship to
disclose but no financial interest!) which is fairly new and presents
the underlying theory in a way that stands on its own even without
supporting Mathematica sofware material:
[Introduction to Probability with Mathematica by Kevin J. Hastings]
http://www.amazon.com/exec/obidos/tg/detail/-/1584881097?v=glance
Of course finite probability spaces tend to be taken up by discrete
mathematics courses, and infinite probability spaces would require for
much generality a treatment of measure theory, so to an extent (as
Rusin observes below) the most accessible portions of undergraduate
probabililty theory tend to get absorbed into a combinatorial
treatment.
For an effort to tackle the measure theory and probability together in
an "undergraduate text", see here:
[Measure, Integral and Probability by Capinski and Kopp]
http://www.mathbook.com/s/Set_Theory/Measure_Integral_and_Probability_Springer_Undergraduate_Mathematics_Series__3540762604.htm
However its strengths seem to lie in the successful integration of the
two subject areas, rather than in its treatment for either of the two
areas on their own merit.
If we were willing to accept something on the merit of being a
Springer-Verlag "undergraduate series" text, then this would fit the
bill:
[The Pleasures of Probability by Richard Isaac]
http://www.amazon.com/exec/obidos/tg/detail/-/038794415X?v=glance
Note however that the book's editorial review seems to reinforce my
suspicion that it is written more for a non-specialist reader than for
an undergraduate course offering.
[Dave Rusin's Known Math: Probability theory and stochastic processes]
http://www.math.niu.edu/~rusin/known-math/index/60-XX.html
Combinatorics
-------------
[A Course in Combinatorics (2nd Ed.) by van Lint and Wilson]
http://www.bookhq.com/compare/0521422604.html
[A Walk through Combinatorics by Miklos Bona]
http://www.wspc.com/books/mathematics/4918.html
[Constructive Combinatorics by Stanton, Stanton, and White]
http://www.azd.com/list/books/0387963472.html
[Dave Rusin's Known Math: Combinatorics]
http://www.math.niu.edu/~rusin/known-math/index/05-XX.html
Integration/measure theory
--------------------------
[Measure Theory and Integration by G. Debarra]
http://www.amazon.com/exec/obidos/ASIN/0470272325/categoricalgeome/104-2438879-0049547
[Lebesgue Measure and Integration: An Introduction by Frank Burk]
http://www.wileyeurope.com/WileyCDA/WileyTitle/productCd-0471179787.html
[A Primer of Lebesgue Integration by H. S. Bear]
http://www.mathbook.com/c/Integrals_Calculus/A_Primer_of_Lebesgue_Integration_0120839709.htm
[Dave Rusin's Known Math: Measure and integration]
http://www.math.niu.edu/~rusin/known-math/index/28-XX.html |