I'd like to know what English-language mathematics textbooks
are used most in the undergraduate university curriculums
for students specializing in mathematics.

For the reason why I want to know this, see
<http://www.cs.kun.nl/~freek/notes/mathstdlib.ps.gz>.

The twelve subjects I'm particularly interested in (these
are the fields that every mathematics student will take
before specializing) are:

  algebra
  analysis
  calculus
  combinatorics
  complex function theory
  differential equations
  geometry
  integration/measure theory
  linear algebra
  mathematical logic
  probability theory
  topology

For each of these subjects I'd like to know at least three
and preferably five textbooks, ranked according to popularity.

I don't particularly consider important exactly where these
books are popular.  If this information is available for
the US universities, or for universities all over the world,
both would be acceptable to me.

This question is pretty important to me: if the answer
doesn't satisfy me I will ask my money back.  On the other
hand if the answer is available but costs more than I'm
offering here, I'd very much like to know.

I'm a fan of mathtalk: if he/she could answer this question
properly, I'd be very happy.

---

Request for Question Clarification by mathtalk-ga on 31 Oct 2003 07:58 PST

Hi, wondering-ga:

Thanks for the kind comments.  I'm "wondering" how fast you need this
information.  Some of the 12 areas are pretty cut and dried as far as
what textbooks are used mode (at American universities), e.g. calculus
and linear algebra (required for essentially all math majors), but
others are going to require some investigation.

For example, mathematical logic is a favorite subject (and I'd love to
have the opportunity to recommend a text written by A. Margaris,
available as a Dover reprint), but you seem interested in sales data
rather than recommendations, or perhaps a count of undergraduate
institutions which have adopted popular texts.  Such numbers will, I'm
afraid, be quite time-consuming to gather.

Let me also say that if a fellow researcher happens to have a source
for the relevant sales or survey information that wondering-ga is
looking for, I'd be pleased to see them step up and answer.

regards, mathtalk-ga

---

Clarification of Question by wondering-ga on 02 Nov 2003 12:29 PST

>I'm "wondering" how fast you need this information.

I'm in no hurry whatsoever.

>but you seem interested in sales data rather than
>recommendations, or perhaps a count of undergraduate
>institutions which have adopted popular texts.

If you've looked at my mathstdlib note, you probably realise
that my goal is to make a stack of twelve textbooks, point
to that stack, and then say "let's formalize those, and we'll
have a proper basis for a library of formalized mathematics."
(Probably put this stack on my webpage somewhere too.)

The reason I'm asking this question on answers.google.com
is that I have no good idea on how to start choosing this
stack in the first place.  Also, because I don't want
people to ask me, after I have chosen twelve books, "have
you looked at book so-and-so, that's one of the standard
textbooks in that subject!" and me then not knowing about it.
So what I _don't_ want is recommendations for good textbooks.
I finished my math studies ages ago, I'm not looking for this
information because of my own studies.  I want to know what 
are "the classics".  Or if a recent textbook is a big hit,
what are "the hits".

I don't need actual sales data (how can you know from that
what's used in university undergraduate courses anyway?) or
actual counts of institutions.  It doesn't need to be
that precise.  Just something that I get the impression
that it's a reasonable approximation of what I'm asking for.

>Such numbers will, I'm afraid, be quite time-consuming
>to gather.

If this means that you want me to raise the price, but
google's rules forbid you to ask for that, just say "yes".

---

Request for Question Clarification by mathtalk-ga on 02 Nov 2003 16:44 PST

Hi, wondering-ga:

Thanks for the prompt clarification.  I appreciate your consideration
of the pricing issue, but rest assured that I would not be reluctant
to suggest an adjustment if I thought that the work required was out
of line with the amount offered.  Mainly I was, before the recent
clarification, confused about the criteria for selection of books.

I read the referenced Web page, and I especially enjoyed the final
observation about the "paradox" that, from an automated proof checking
perspective, some of what is treated as "elementary" in the classroom
is actually "dependent" on the "advanced" material.  As you proceed
with this project I think you will encounter this "inversion" of
progress quite often.

The Postscipt note (which is not gzipped, despite the file extension)
presents three (partly conflicting) criteria for inclusion:

"Here are the criteria that such a textbook has to satisfy:

-  It has to be rather thorough. That is, it should present all of the
theory, including the more basic results in the area.

-  It has to be standard. It's far more important that the book
follows accepted practice in the field than that it's mathematically
elegant.

-  It should not be too extensive (to keep things managable.) So it
shouldn't as much be a reference to the subject [as] a text meant for
teaching."

The sense I have now of your request is that you'd like to know, of
the several books in each field, which is considered a "classic",
which a "recent hit", and which is considered especially
representative of teaching practices.  Often these criteria will not
be met by a single book, and perhaps this explains your interest in
having three to five books identified in each of the twelve fields.

Since you are not in any great hurry, I will have the luxury of
pondering before contributing at least some Comments in specific
areas.  Other Researchers are of course welcome to participate, and
I'm leaving the Question unlocked in case someone has a sufficiently
strong interest to take the lead.

regards, mathtalk-ga

---

Clarification of Question by wondering-ga on 03 Nov 2003 04:38 PST

>(which is not gzipped, despite the file extension)

The file in my web directories is gzipped, and when I ask for
it with wget I get a gzipped file as well.  Someone must be
uncompressing it on the way to you (but not our web server
I guess, because else my wget would not get a gzipped file?)

>Often these criteria will not be met by a single book,
>and perhaps this explains your interest in having three
>to five books identified in each of the twelve fields.

This is correct.  I think that if you could indicate for each
field just one book that reasonably satisfied the
three criteria from my note (the ones you quote), then that
already would be very interesting to me.  (I have no real
ideas about textbooks myself: when I studied math long
ago, I learned everything from stenciled course notes.)
But of course I'd rather make the final selection of twelve
textbooks myself, according to my own tastes.

>Since you are not in any great hurry, I will have the
>luxury of pondering before contributing at least some
>Comments in specific areas.

That would be great!  But of course I would even more
appreciate a full answer :-)

---

Request for Question Clarification by mathtalk-ga on 26 Nov 2003 17:20 PST

Hi, wondering-ga:

Just to let you know, I plan to post an Answer within 24 hours.  The
one area that is least clear to me is Geometry.  If you can clarify
which kind of Geometry you have in mind, that would be great.

In my experience Geometry would most likely be a topic in a Math
Education curriculum, in the sense of exposure to teaching Euclidean
and non-Euclidean topics to illustrate an axiomatic method.  Of course
Analytic Geometry is somewhat part and parcel of the Calculus
curriculum, but in itself would have to be considered at least partly
remedial at the college level (notwithstanding its extensive overlap
into multi-variable calculus and linear algebra).

In days gone by one might expect a substantial exposure of
undergraduates to Algebraic Geometry, and this may still be the case
in some departments.  Let me know if this is what was intended.

regards, mathtalk-ga

---

Clarification of Question by wondering-ga on 28 Nov 2003 14:22 PST

Hi mathtalk-ga,

>Just to let you know, I plan to post an Answer within
>24 hours.

Great!  (I had feared you had forgotten me :-))

>The one area that is least clear to me is Geometry.
>If you can clarify which kind of Geometry you have in mind,
>that would be great.

I got my list of subjects by comparing the math programs
of the Dutch universities.  So I had the kind of geometry
in mind that they teach in the undergraduate courses here.

>In my experience Geometry would most likely be a topic in
>a Math Education curriculum, in the sense of exposure to
>teaching Euclidean and non-Euclidean topics to illustrate
>an axiomatic method.

I think the Dutch universities are not significantly
different from the US ones (I guess those are what you're
talking about) in that respect then.

>Of course Analytic Geometry is somewhat part and parcel of
>the Calculus curriculum, but in itself would have to be
>considered at least partly remedial at the college level
>(notwithstanding its extensive overlap into multi-variable
>calculus and linear algebra).

Yes.  But I don't think that courses about this would be
labeled "geometry" in a Dutch curriculum.

>In days gone by one might expect a substantial exposure of
>undergraduates to Algebraic Geometry, and this may still
>be the case in some departments.  Let me know if this is
>what was intended.

Not really, I don't think undergraduates in the Netherlands
get much algebraic geometry.  (In fact even later I haven't
ever seen much algebraic geometry, much to my regret.)

I think what I also have in mind when I think of a course
in geometry, is what I once (long long ago) got as part
of the training for the mathmatical olympiads (the Dutch
organiser of that was specialized in geometry, and this
had repercussions on what we were taught).  It went on in
great detail about things like affine geometry, projective
geometry, inversive geometry, etc.  With proofs that the
class of transformations that map lines into lines are
exactly the affine transformations (?), the ones that map
lines/circles into lines/circles are exactly the inversive
transformations, etc.

I also think there was stuff on things like what "length",
"angle", "area" etc. numerically means in non-Euclidean
geometries.  I seem to remember tables with goniometric
expressions in them describing this analytically, with each
time three variants of everything: for elliptic, "normal"
(parabolic) and hyperbolic geometry.  Formulas like how the
area of a triangle was related to the excess of the sum
of the angles of the triangle in hyperbolic and elliptic
geometry and so on.  I don't remember the details.

So that's what in my mind too as the subject matter of
"a course of geometry".  But that's personal, it's not the
original spec of what I was looking for.

I'd say: decide for yourself how to interpret what is
undergraduate geometry.  I'm already very happy to get an
answer from you after all!

Many regards back, and thank you already,
wondering-ga

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

---

Request for Answer Clarification by wondering-ga on 07 Dec 2003 13:03 PST

Dear mathtalk-ga,

>I'll supply those remarks on Combinatorics and 
>Measure/integration theory tomorrow.

Just reminding you of this promise :-)

Sincerely, wondering-ga

---

Clarification of Answer by mathtalk-ga on 11 Dec 2003 17:52 PST

The combinatorics subject matter and theory of integration/measure
theory are both closely related to other undergraduate courses, and
it's not clear that there is any "standard" textbook for
undergraduates in the United States.

The recommended selection for combinatorics is one of my favorite
"pasttime reading" texts.  I own two copies!  Van Lint and Wilson
prepared the text from lecture notes used in a couple of (advanced)
undergraduate seminars, but in the scheme of things it comes closer
than any of my other recommendations to being a "comprehensive"
reference book rather than an easily polished textbook.

But in relationship to the subject matter of combinatorics one cannot
really say it is comprehensive, merely that it is written at an
introductory level but with coverage than could be usefully presented
across two semester classes (which would be an unusual allocation of
time to this material at most American colleges).

The back up recommendations are less extensive, and were chosen to
present a couple of clear alternatives.  The book by Miklos Bona aims
to provide undergraduates with a succinct but intelligent introduction
to the methods and subject matter of combinatorics.  This is itself a
challenge, because there is scarcely any identifiable subject in
mathematics with less coherence of method.  I refer to combinatorics
as the "crazy uncle" of the mathematics world.  The results there may
strike you one moment of portending great depth; another moment as
being completely accidental.

Combinatorics, which is largely the art of enumeration, encroaches on
many other disciplines in mathematics (notably number theory and
algorithmic graph theory, but even these are not broadly overlapping
with it).  The third selection provides a focus on constructive
(algorithmic) aspects of combinatorics, which might make it suitable
for a discrete mathematics specialization.

When I came to the last subject, integration/measure theory, I was
forced to speak without any personal knowledge of how this subect
would be taught to undergraduates.  I've ranked the selections based
on how others have described them.  As an undergraduate student I was
introduced to the Riemann and Riemann-Stietljes integrals; measure
theory appeared in the context of probability spaces but the subtlety
of Lesbesgue theory was reserved for 1st year graduate courses.

The book by DeBarra was most often presented as an undergraduate text
in this subject.  The other two are described as "introductory" and
"primers" but I'm not certain if this would really make them suitable
as undergraduate textbooks.

Thanks for posting this challenging question; it stirred a lot of
thought about the structure of undergraduate mathematics education. 
I'm not sure that the rest of your project will flow smoothly from
this selections, as the attempt to "formalize" mathematical results in
the undergraduate curriculum is often a matter of polishing the less
regular structure, useful to those who will either be satisfied with
pragmatic calculational abilities or go on to a more complete
understanding in graduate school.

regards, mathtalk-ga

Dear mathtalk-ga,

Thank you VERY much for answering my question!

So now I have the pleasurable task to go look at all 
those books.  I hope I can find most of them!

Sincerely,
wondering-ga

Hello Matey, good luck with the course, here's some English Maths
Books from an English University's Maths Course's Reading List:
Have a look
http://www.city.ac.uk/mathematics/readlist.pdf
I'm sure you'll agree that answers your question
Ben

If you are a hardcore undergrad student you will probably know most of
calculus when you start your university studies, here are some good
books though suitable for good undergrad students.

  abstract algebra
    first "abstract algebra by herstein"
    after that "topics in algebra by herstein"
  analysis 
    certainly "principles of mathematical analysis by walter rudin"
  complex function theory 
    "complex analysis by ahlfors"
  linear algebra 
    "linear algebra by serge lang"
  topology 
    perhaps "topology by munkres"


Regards,
/m