# Possible to learn Calculus from an Analysis book?

So I'm trying to decide which of the following to get:

Spivak (Calculus)
Courant (Introduction to Calculus and Analysis)

I've heard both of these are analysis books. I'd like to try and self teach myself using one of them, until I take calc 1 + 2 in the summer. I'd also like a more rigorous approach to math to give myself a deeper understanding of it. My question is, can I learn calculus from the ground up from an 'analysis' book? I'm really not sure of the difference between calculus and analysis.

jbunniii
Homework Helper
Gold Member
It is an exaggeration to call either of these an analysis book. A better characterization would be that they are rigorous calculus books.

Courant covers more applications than Spivak, and is just a little bit looser with the rigor.

Spivak dots every i and crosses every t in his proofs and covers limiting/epsilon-delta arguments in much more detail, whereas I thought Courant's treatment was more rushed and brusque.

Spivak's exercises emphasize proofs whereas Courant's emphasize applications.

Both are excellent books, and if possible you should take a look at both of them so you can decide for yourself which suits you better.

You can certainly learn calculus from either of these books. They are both much more demanding than the typical calculus book, but if you are up to the challenge then you will come away much better equipped.

There isn't a well-defined boundary between calculus and analysis. Both deal with limiting processes in the real number system, in particular differentiation and integration.

"Calculus" often carries the connotation of emphasizing how to calculate derivatives and integrals, whereas "analysis" usually implies treating the same topics with greater rigor, more generality, and more emphasis on theoretical questions versus calculations or applications. But there's a lot of overlap.

I'm mostly through Spivak as my first exposure to calculus. If your algebra is solid, and you're very thorough in studying the book (laboring over every detail in the chapters), then you'll get a better calculus education than a more watered down approach. You can compensate for less applications by picking out good problem books, I think.

It also would help a lot to brush up on how proofs work.

I think it's more beneficial to take the following approach -

"problem, problem, and problem", then introduce to proofs/logic, and then finally revisit using proofs/logic.

start from top one at a time -

1) Calculus: Early Transcendentals <- pretty good when it comes to problem solving

2) An Introduction to Mathematical Reasoning <- will build you a fantastic proof/logic base

3) Real Analysis <- alike other springer books, very easy to read and might i add it gets the job done well

4) Understanding Analysis <- further insight