Calculus (ELEMENTARY)
Of course, as we all know, the One True Calculus Book is
Spivak, Calculus
This is a book everyone should read. If you don't know calculus and
have the time, read it and do all the exercises. Parts 1 and 2 are
where I finally learned what a limit was, after three years of
bad-calculus-book “explanations”. The whole thing is the most
coherently envisioned and explained treatment of one-variable calculus
I've seen (you can see throughout that Spivak has a vision of what
he's trying to teach).
The book has flaws, of course. The exercises get a little monotonous
because Spivak has a few tricks he likes to use repeatedly, and
perhaps too few of them deal with applications (but you can find that
kind of exercise in any book). Also, he sometimes avoids
sophistication at the expense of clarity, as in the proofs of Three
Hard Theorems in chapter 8 (where a lot of epsilon-pushing takes the
place of the words “compact” and “connected”). Nevertheless, this is
the best calculus book overall, and I've seen it do a wonderful job of
brain rectification on many people.
[PC] Yes, it's good, although perhaps more of the affection comes from
more advanced students who flip back through it? Most of my exposure
to this book comes from tutoring and grading for 161, but I seriously
believe that working as many problems as possible (it must be
acknowledged that many of them are difficult for first year students,
and a few of them are really hard!) is invaluable for developing the
mathematical maturity and epsilonic technique that no math major
should be without.
Other calculus books worthy of note, and why:
Spivak, The hitchhiker's guide to calculus
Just what the title says. I haven't read it, but a lot of 130s
students love it.
Hardy, A course of pure mathematics
Courant, Differential and integral calculus
These two are for “culture”. They are classic treatments of the
calculus, from back when a math book was rigorous, period. Hardy
focuses more on conceptual elegance and development (beginning by
building up R). Courant goes further into applications than is usual
(including as much about Fourier analysis as you can do without
Lebesgue integration). They're old, and old books are hard to read,
but usually worth it. (Remember what Abel said about reading the
masters and not the pupils!)
Apostol, Calculus
This is “the other” modern rigorous calculus text. Reads like an
upper-level text: lemma-theorem-proof-corollary. Dry but comprehensive
(the second volume includes multivariable calculus).
Janusz, Calculus
The worst calculus book ever written. This was the 150s text in
1994–95; it tries to give a Spivak-style rigorous presentation in
colorful mainstream-calculus-book format and reading level. Horrible.
Take a look at it to see how badly written a mathematics book can be.
