Recommendation about studying calculus.

Question

I will be first year undergrad in Physics next year. I've studied calculus from stewart so far and finished it until the integral part. Now, I am studying calculus from Apostol, because it starts with integral and my question will be about this. Should I continue with Apostol or study from the books like Adam's, Stewart etc.?

Answer 1

I think it's good to study calculus rigorously if one wants to go far in physics. The kind of thinking using inequalities that goes into proofs in analysis - such as "this term eventually becomes less than $\varepsilon$ for large $N$" - is of constant use in physics. It provides the firm foundations for when a physics professor waves his hands and says "this term is negligible" - it's what he knows because he's learned math properly, but doesn't necessarily say out loud in talking to lower-level undergraduates.

Apostol's book also gives you a level of technical proficiency in computations that you simply can't get in an average textbook. The theory and technique go hand in hand. Apostol's book is also full of examples of applications to the sciences. It was used as the main calculus textbook at Caltech for many years, including for engineering students.

The main drawback to Apostol is that there is no solution manual, as far as I know. Spivak's book has a complete solution manual. The problem is that Spivak's book is somewhat more theory-centred, including topics of greater interest to mathematicians (such as proving the existence of a continuous, nowhere differentiable function). Overall, you might find the issue of the solution manual more important, however.

You should also realize that in practice, integrals are usually calculated by using derivatives. This connection can obviously only be made in Apostol once he's also studied derivatives, and he does it in Chapter 5. On the other hand, the first two chapters will give you a very good idea of what an integral is - much better than Stewart of course, although about ten times more work too.