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.
Best Answer
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:
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.
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.