I'm interested in self-studying the following books over the next year or so:

  1. Spivak's Calculus (I'm already in Ch. 5 and it is very slow going)
  2. The Cauchy-Schwarz Master Class by J. Michael Steele
  3. Analytic Inequalities by Nicholas Kazarinoff

My goal in studying these books is to gain a deeper understanding of calculus, basic real analysis, and manipulations of the standard inequalities, with the ultimate goal of understanding derivations, approximations, and inequalities in probability and statistics (Stirling's approximation, Wallis product, Gamma Function, Normal Distribution, Limit Theorems etc). One of the things I realized when I first started studying Spivak's Calculus is that I have had very little experience in solving challenging problems. I have never had any issues with doing 'Exercises' in the standard engineering style calculus text books, but I am often at a loss of ideas when I do problems in Spivak.

My questions are the following:

  1. Before progressing through Spivak, should I go through a book like Art and Craft of Problem Solving by Paul Zeitz? I guess the point of doing this would be to beef up my problem-solving skills. I should note that I am not very excited about working through the Art and Craft of Problem Solving because a lot of it seems geared toward solving Olympiad geometry problems. I've never had a solid geometry course, so at this point I feel like it might just be a waste of time trying to learn plane geometry.

  2. Should I relearn high school mathematics? To be perfectly honest, I feel robbed by my entire education and I'm very disappointed by my lack of foresight up to this point. I've always used easy textbooks(not my choice) in my college calculus, Linear algebra, and ODE and PDE classes and believed 'good grades' were enough.

  3. Or, should I just keep a copy of Polya's Heuristics on hand while I patiently work through Spivak?

I'm just looking for a bit of advise on the wisest way to proceed.


  • 3
    $\begingroup$ I wonder how modern students and professors miss the much better book "A Course of Pure Mathematics" by G H Hardy which is especially designed for self study and for students of young age (16-17 years). And I also don't understand the reason of extreme popularity of Spivak's Calculus. In terms of content "Spivak Calculus = Hardy Pure Math + proof that $\pi$ is irrational + proof that $e$ is transcendental" and in terms of quality "Hardy Pure Math >> Spivak Calculus". $\endgroup$
    – Paramanand Singh
    Commented Jun 22, 2014 at 7:24
  • $\begingroup$ @ParamanandSingh Really? Can you elaborate on what aspects of Hardy are such high quality vs Spivak? I've only studied each a little, but find Spivak very quickly motivates deep rigor; has clear, fun, and fast exposition; and great problems. $\endgroup$ Commented Jun 9, 2023 at 9:12
  • $\begingroup$ @SRobertJames: well the language and presentation in Hardy's book is so motivating. You never feel like taking some course but rather you are on an exciting journey to discover completely new stuff. This is especially when you are a calculus beginner and your math knowledge consists only of basic algebra and trigonometry. I once wrote a bit personal review of the book here: paramanand.blogspot.com/2005/11/… $\endgroup$
    – Paramanand Singh
    Commented Jun 9, 2023 at 11:14

2 Answers 2

  • I'd suggest that you check out Daniel Velleman's book: How to Prove It, particularly if you'd like to develop more insight into how to approach proofs in a wide range of contexts.

    The table of contents of the book are available at the link above; just click on the text/arrow "Look inside!" (located just above the image of the book).

  • See also Polya and Conway's book How to Solve It: A New Aspect of Mathematical Method


I will give you advice on Spivak but realize that it applies to almost every math book in general. When I first worked through the book it was my first experience with analysis or "advanced calculus" and the first few chapters took me the longest time to get through. It takes a few chapters to learn the author's style. After the first few chapters Spivak becomes much easier to get through. The "breaking point" for me was when derivatives were introduced. After this, I knew what to expect with the exercises and I knew how to deal with the types of problems he assigned.

I worked through another book with inequalities that included a new notation which saved a lot of space at the expense of clarity. The text was also filled with tiny mistakes. After the first two chapters I learned to deal with the notation and how to get through the text without relying on the author's work.

Keep working through Spivak for now. It gets easier. If you take an easy way out you are at risk of getting stuck again when you have to work through Rudin or other high level analysis books. I would also recommend working through another linear algebra text which shouldn't be too difficult if you have already seen it before.

I don't usually recommend olympiad style books unless you want to get better at olympiad style mathematics. If you want to get better at Spivak type problems you need to work Spivak type problems. Books like How to Solve it would only provide a marginal benefit at best.


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