My professor of calculus said that he will not adopt books like "stewart" or "thomas" because they are too easy for a physics undergrad course.

He said that Apostol is a great book for our case, but some people told me that this book focus a lot in the historical concept and was not good to them.

I know that my professor likes extreme rigour in math, so do i.

In the words of Apostol itself (preface):

Some people insist that the only way to really understand calculus is to start off with a thorough treatment of the real-number system and develop the subject step by step in a logical and rigorous fashion

What books follow this principle of learning calculus?

Thank you.

  • 1
    $\begingroup$ A Primer of Infinitesimal Analysis, John Bell; since you're doing physics. $\endgroup$
    – user117644
    Mar 12, 2015 at 7:32

4 Answers 4


Spivak's Calculus and Calculus on Manifolds (for multivariable calculus) are pretty standard rigorous calculus texts. Rudin's Principle's of Mathematical Analysis is standard for a first analysis course, but may be too abstract for a physics course. If you don't like Apostol but still want the mathematical rigor, these are good alternatives.

There's also Strang's free calculus text here.

For more calculations, Hubbard's Vector Calculus book works as well.

Edit: Yes, it does depend on where you are from, since American students (such as myself) wouldn't study Spivak in our first year (to be fair, I actually never studied Spivak's Calculus, just his Calculus on Manifolds for a freshman advanced math course but this was far from the norm and only for math majors, not physics majors) and it is considered to be one of the best rigorous calculus textbooks as far as I'm aware. Seeing as the OP's professor mentions Apostol (although not mentioning which Apostol book he means, either his two Calculus books or his more advanced Mathematical Analysis), I think Spivak's Calculus books are a good alternative to Apostol's Calculus books, although they do not contain any differential equations theory or as extensive a treatment of linear algebra as Apostol (Calculus on Manifolds includes a bit of linear algebra in as far as it allows him to discuss total derivatives and multilinear forms).

All this said, I'd honestly need a bit more information on the course in order to give a proper alternative. There are far too many advanced "rigorous" vector calculus textbooks to choose from for such a general recommendation, so I gave what are considered by most people to be fairly standard references.

There are other books of course:

  • Fitzpatrick's Advanced Calculus: Again, little in the way of
    linear algebra or differential equations, but I think he explains
    integration much better than Spivak and a bit more drawn out explanations, which I think is nice if you're new to the material.
  • Boas's Mathematical Methods in the Physical Sciences: This book probably has more material than you need (PDEs, integral transforms, basic complex variables). However, I'd say probably the first 400 pages is a great alternative to Apostol, and as the title suggests it's good for a physics course.
  • Stroyan's Foundations of Infinitesimal Calculus: Haven't read this one to be honest, but seems pretty good based on the table of contents, although it does seem a bit short.
  • Protter's Basic Elements of Real Analysis: A nice springer undergraduate text which is an abstract version of Stewart's Calculus, but not quite as rigorous as others on this list. For more rigor but using a similar style, he also has a book called A First Course in Real Analysis, which I also like.

These are my best "general recommendations". Honestly, we do need me information to give good recommendations (does your class have a website/what is the class on/etc.).

Hope this helps.

  • 1
    $\begingroup$ In Italy, a book like Spivak's Calculus would be considered extremely easy for undergraduate level. $\endgroup$
    – Lonidard
    Mar 12, 2015 at 7:19
  • $\begingroup$ @bharb What book do you recommend for my case? $\endgroup$
    – Voyager
    Mar 12, 2015 at 9:16
  • 1
    $\begingroup$ @bharb American math students (though I don't know that the OP is American) start university a couple of years behind European ones; they've almost never seen anything anywhere close to the level of Spivak. $\endgroup$ Mar 25, 2015 at 17:16
  • 1
    $\begingroup$ @bharb That makes sense. Would you even take a "calculus" course in university in Italy? I would have expected "analysis" from the first year. Though I have no idea whether a clear distinction even exists in Italian. $\endgroup$ Mar 25, 2015 at 20:47
  • 1
    $\begingroup$ @Voyager Edited to add more references that you may be interested in. $\endgroup$
    – Moya
    Mar 25, 2015 at 21:26

Depending on the course, you could use Spivak as Moya mentioned. I'm a fan of Calculus Deconstructed by Nitecki for single variable.


Arihant Publishers Calculus by Amit M Aggarwal is the best as per your requirements.

This is from where i learnt https://www.youtube.com/results?search_query=profrobbob+calculus+2..im sure it will help!


You probably mean "math" undergrad course. In your case, you should check "About Vector by Hoffman" and "div grad curl and all that by schey." But of them are cheap books, and they are pretty abstract, but they are only use for Cal 3. From cal 1-2, I used steward, which is pretty basic though.


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