calculusintegrationreal-analysis
I am currently a second year undergraduate student in mathematics; I am taking Integral Calculus, however I have run into a difficulty.
This is the discomfort that some books of integral calculus generate in me, for the moment I am using Spivak's book of infinitesimal calculus, however I want to know which books you recommend me to strengthen my learning of integral calculus. This taking into account that the book should be focused on mathematicians.
Thanks
I have read through the first few chapters of Spivak, however my personal preference is for Apostol's Calculus. It's also a very rigorous approach, and a very well respected book, however it starts more gently than Spivak's. With Spivak's book, the problems start out extremely hard, and get easier as the book goes on (mostly by getting used to his style, not objectively). With Apostol I was able to understand and answer all the questions in the first few chapters much more easily, and then I saw the difficulty increase a bit; however it increases progressively throughout the book. Many of the problems in the introduction of Apostol are exactly the same as those in Spivak, however the order and context that they are presented in leads you to the correct method for proving them, whereas Spivak's are more isolated.
There are many great discussions about calculus books on other forums, such as The Should I Become a Mathematician? Thread on Physics Forums.
I agree wholeheartedly with mathwonk's statement that, although the books are difficult, reading different approaches and going over them multiple times is really what gives you a deeper understanding of calculus. Mathwonk also mentions that most students find Apostol very dry and scholarly, where Spivak is more fun; however, I have not found this to be the case. I have worked through every problem in Apostol's Calculus through chapter 10 so far, and it has been a joy (most times). As an added bonus, Apostol's Calculus covers linear algebra as well, and the second volume covers multivariable calculus. Spivak's analogous book, "Calculus on Manifolds", is known as an extremely difficult text, and is commonly used as an introduction to differential geometry (indeed, his comprehensive volumes on differential geometry mention Calculus on Manifolds as a prerequisite).
The choice of book should also reflect your future interests. I am a computer programmer currently, and am looking to go into mathematics exclusively. It sounds like you are still melding the two. I would say that Apostol's book might serve you a little better in this respect as well, as it is slightly tilted towards analysis, whereas Spivak's is tilted towards differential geometry. For instance, Apostol introduces "little-o" notation, a cousin of "big-O" notation which is used extensively in computer science. That being said, Spivak has been described by some as a deep real-analysis text more than a calculus book, so you would still deeply cover all the fundamentals.
Another set of calculus books which I own and are held in high regard are Courant's. My brief skim of them, as well as other's comments, suggest that they are more focused on applications perhaps than some of the other books. Apostol's is still, in my opinion, very well peppered throughout with applications; many chapters contain a specific "applications of ..." section which links the theoretical concepts you just learned with the applied use of those concepts.
My only exposure to Courant's expository style comes from his excellent book What is Mathematics. This is a book I would strongly recommend reading regardless of what calculus book you choose. I cannot praise Courant's lucid writing highly enough, and look forward to working though his Calculus texts in the future.
I think that you would find Apostol's book sufficiently rigorous, as well as extremely intuitive. I also am a musician, and coupled with my computer programming experience it seems that perhaps we think alike. Whatever book you choose, recognize before you start it that you are running a marathon, not a sprint.
If you're concerned about time, I don't think reading Calculus by Spivak is the best thing to do.
Either Zorich or Apostol is a great choice. I would say that they're "intermediate" in difficulty. Zorich contains more in-depth discussions of topics, and more examples than does Apostol.
If your goal is only to move on to Royden, you'll probably cover the material more quickly in Apostol. Zorich covers a number of topics not addressed in Apostol, such as vector analysis and submanifolds of $\mathbf{R}^n$. These are important topics, but not direct prerequisites for Royden. Still, I think with all the Lagrange multipliers and similar tools people use in economics, the submanifold topic is important if you want to understand the theory very clearly.
Zorich's first volume is quite concrete, whereas Apostol becomes abstract more quickly. This is probably because he doesn't want to duplicate what would be in a rigorous calculus book like his Calculus, although he does this more than Rudin's book does. His analysis book was for second- or third-year North American students, whereas Zorich's is, at the outset, for first-year Russian ones. Russian students have typically had some calculus in high school, but the practical portion of learning calculus continues into their first-year of university, with harder problems. So in Zorich I, you deal with hard problems on real numbers, rather than delving straight into metric spaces as you would in Apsotol's book.
Zorich covers only Riemann integration, whereas Apostol has chapters on Riemann-Stieltjes integration in one variable, Lebesgue integrals on the line, multiple Riemann integrals, and multiple Lebesgue integrals. The treatment of Lebesgue integration is less abstract than in more advanced books. Since it's limited to $\mathbf{R}$ or $\mathbf{R}^n$, it's more elementary, but at the same time there is some loss in clarity compared to the abstract theory on measure spaces. One reason to use Apostol might be a sort of introduction to the Lebesgue theory before returning to it at a higher level and "relearning" certain parts of it. Whether you'd want this is up to you.
The fact that both Rudin and Apostol have chapters on Riemann-Stieltjes, rather than Riemann, integration, indicates to me that they assumed students had already studied Riemann integrals rigorously, and would be ready for a generalized version right from the start. Considering the type of calculus courses most students take these days, this is rarely the case now. Zorich doesn't have this problem.
All in all, for a typical student who is good at math but didn't learn their calculus from a book like Spivak's or Apostol's Calculus, I think Zorich is the better choice because of the more concrete approach in the first volume (this doesn't necessarily mean easier). On the other hand, time constraints might cause you to prefer Apostol's analysis book.
EDIT: An important point that I neglected to mention is that Zorich's book will be much better than Apostol's if you aren't yet acquainted at all with multivariable calculus. A practical knowledge of some multivariable calculus is probably one of the tacit assumptions that Apostol and Rudin make about their readers, which is what allows them to deal with multivariable calculus in a briefer and more abstract way. Compare Apostol's 23-page chapter on multivariable differential calculus to Zorich's 132 (in the Russian version).
EDIT: Based on your later comments, I would suggest that reading
Spivak's Calculus,
Whichever you prefer of Apostol's Mathematical Analysis or Rudin's Principles of Mathematical Analysis.
would be a reasonable plan.
However, before beginning the multivariable calculus parts of those books, it would be best to learn some linear algebra and multivariable calculus from another source. This could be Volume 2 of Apostol's Calculus. You could instead skip straight to the multivariable part of Volume 1 of Zorich, but you'd have to learn the necessary linear algebra elsewhere first. I don't recommend Spivak's Calculus on Manifolds if you want to learn multivariable calculus for the first time. Also, you won't need Munkres - you'll get enough topology to start with in whichever other book you read.
EDIT: In answer to your additional question, these topics are mostly not discussed in Spivak.
However, Spivak is an excellent introduction to the mathematical way of thinking. That is, although you will not learn all the specific facts that arise in higher-level books (you do learn many, of course), you will learn to read and understand definitions, theorems and proofs the way mathematicians do, and to produce your own proofs. You will become intimately familiar with real numbers, sequences of real numbers, functions of a real variable and limits, so you will have examples in mind for the more general structures introduced in topology. You will also solve difficult problems.
So it is not that you will know topology already when you've read Spivak's book, it's mainly that it ought to be easier for you to learn because you will have improved your way of approaching mathematical questions. Countable sets are in fact discussed in the exercises to Spivak, however.
I can't guarantee that your trouble will "go away," but there is a good chance it will.
Also feel free to use Zorich rather than Rudin or Apostol, after Spivak, or even to jump straight to the multivariable part of Zorich at the end of Volume 1 and start reading from there.
Best Answer
So this is my view on calculus books. The best book on calculus is Fichtenholz, Differential und Integralrechnung. It is 3volumes. The original is russian, and it has been translated into german and polish. It is solid and rigorous exposition of all issues in calculus with many examples.
Another classic book is Joseph Edwards, Treatise on Integral Calculus 2vols. It is in english and can be found online on internet archive. It is very extensive on examples and methods of integration, but it does not cover theoretical aspects. He also has Integral Calculus for Beginners, a smaller and more approachable book. And has books on Differential calculus.
Another traditional text is Todhunter, Treatise on Integral Calculus, and Treatise on Differential Calculus, less extensive than Edwards, but easier to assimilate.
The modern text for rigorous calulus is Rudin, Principles or Real Analysis. Solid proofs of the theory, although using topological concepts, but weak on examples.
Another good book is Philip Franklin, Treatise on Advanced Calculus.
Looking at your example, it is more of a practical question and not so much theoretical, I recommend Todhunter, or Edwards, Integral Calculus for Beginners whichever you find better.
Of course, this list betrays my preference for older books. Finally, one cannot become an expert in any subject by reading one book no matter how good it is, you must read several.