well i like em both very well. apostol as i recall is more precise, having been written 30 years later, in the era of careful mathematical writing, the 1960's, whereas courant was a pioneering text from the 30's and is by a guy who is also a writer on mathematical physics. so it has more applications. but i think apostol has some of those too. apostol is more meticulous and dry, courant has more charm.
so courant is not by modern standards as "rigorous" as apostol, but the essentials are there.
apostol is incredibly scholarly and careful, and clear too. both of them start with integration instead of differentiation, the correct historical ordering.
gosh I cannot pick one over the other, but I'm leaning towards courant.
to be honest though, as a young student, courant was hard for me to get far in.
i did not realize i should read it in palatable pieces, and got discouraged by not making it through the dense parts.
apostol can do that to you too.
as a teacher though, i was amazed at how carefully apostol covered all the bases.
i myself have seldom learned anything straight out of one book, at one session. everything has to go around and around for me, and settle out, in ways that are partly psychological.
so I use more than one book.
young people really like spivak though. he works hard to make it appeal to the very bright, but also naive young student.
i actually learned much of calculus for the first time out of spivak, while grading the course as a grad student.
then years later as a teacher i realized that the same stuff was in courant, and that spivak had just cleaned it up, and repackaged it.
then more years later, i read apostol and was again impressed that i was still learning a lot i still did not know.
you cannot go wrong with any of those books. but don't feel left out if they do not suit at first encounter.
shoot, my best math buddy at harvard, taking the elite honors calc class said his favorite book was silvanus p thompson's calculus made easy, and i like it too.
it took me years to realize that the down home funky stuff thompson says is actually right though, because he didn't prove anything. i like proofs.
stewart can be useful too, and thomas, or edwards and penney. i sniff at some of these as cookbopoks, but I am willing to learn anywhere i can.
i get a little out of each source, whatever that source does well.
there is no oine source in general that does everything best.
(I might make a one or two exceptions in my own very specialized field of research, as there are a couple of experts who have written some great texts on things they excel everyone at.)
but the perfect calculus book does not exist.
I like Spivak, because it spoke clearly to me, and allowed me to make the amterial my own, so that I no longer need to refer to the book. When someone gets beyond a book, and begins to pooh the very book he used to value, that isa big compliment to that book. i.e. the book gave him all he needed and allowed to amke it his own, so that he no longer needs the book. then the book falls away and becomes superfluous. that is a good book.
spivak did that for me.
but i got some ideas from courant and some from apostol.
So it is one thing to say how scholarly a book appears to an old person, and another to say which one a young person can best learn from. that is a personal matter. One of my friends however who is a professor of topology learned very well from apostol as a student at MIT. Spivak was used at Chicago recently, and Courant was used in the 60's at Harvard.
I myself own them all, as well as Joseph Kitchen's fine book, and also G.H Hardy's classic, Pure Mathematics, and also Jean Dieudonne's Foundations of Modern Analysis, and also Lynn Loomis' Advanced Calculus, and also Serge Lang's Analysis I and II. I especially like Lang's books. They are so clear, and to the point.
I also own Calculus Made Easy, Stewart, Edwards Penney (6 different editions), Lipman Bers book, Thomas (several editions), and am recently trying to incorporate the ideas of lebesgue integration into my grasp of calculus, from books like Rudin's Real and Complex Analysis, and Riesz Nagy's Functional Analysis.
