Writing a calculus book that's more rigorous than typical books is hard because if you go too hard, people will say that you've written a real analysis book and the point of calculus is to introduce certain concepts without going full analysis. This book seems to have at least avoided the trap of trying to be too rigorous about the concept of convergence and spending more time on introducing vocabulary to talk about functions and talking about intersections with linear algebra.
As a math professor who has taught calculus many times, I'd say there are many different things one could hope to learn from a calculus course. I don't think the subject distills well to a single point.
One unusual feature of calculus is that it's much easier to understand at a non-rigorous level than at a rigorous level. I wouldn't say this is true of all of math. For example, if you want to understand why the quadratic formula is true, an informal explanation and a rigorous proof would amount to approximately the same thing.
But, when teaching or learning calculus, if you're willing to say that "the derivative is the instantaneous rate of change of a function", treat dy/dx as the fraction which it looks like (the chain rule gets a lot easier to explain!), and so on, you can make a lot of progress.
In my opinion, the issue with most calculus books is that they don't commit to a rigorous or to a non-rigorous approach. They are usually organized around a rigorous approach to the subject, but then watered down a lot -- in anticipation that most of the audience won't care about the rigor.
I believe it's best to choose a lane and stick to it. Whether that's rigorous or non-rigorous depends on your tastes and interests as a learner. This book won't be for everybody, but I'd call that a strength rather than a weakness.
The rigorous form of the non-rigorous version is non-standard analysis: There really are tiny little numbers we can manipulate algebraically and we don't need the epsilon-delta machinery to do "real math". It's so commonsensical that both Newton and Leibniz invented it in that form before rigor became the fashion, and the textbook "Calculus Made Easy" was doing it that way in 1910, a half-century before Robinson came along and showed us it was rigorous all along.
> The rigorous form of the non-rigorous version is non-standard analysis
This is quite overstated. There are other approaches to infinitesimals such as synthetic differential geometry (SDG aka. smooth infinitesimal analysis) that are probably more intuitive in some ways and less so in others. SDG infinitesimals lose the ordering of hyperreals in non-standard analysis and force you to use some non-classical logic (intuitively, smooth infinitesimals are "neither equal nor non-equal to 0", wherein classical reasoning would conflate every infinitesimal with 0), but in return you gain nilpotency (d^n = 0 for any infinitesimal d) which is often regarded as a desirable feature in informal reasoning.
One of the dangers of a non rigorous approach is not being clear about relative rates. If you're not being precise you're going to confuse people when you say eg that in the limit this triangle is a right triangle. Or look at Taylor's theorem. In different limits you can say a curve is a line, a parabola, a cubic, etc.
Anyway you've already got Apostol - if it's just calculus as such get an older edition. Modern ones have extra goodies like linear algebra but have modern text book pricing (cries softly in $150/volume).
Getting an old enough edition of Apostol's "Calculus" to not include linear algebra might be a bit challenging. Linear algebra was added to both volumes in their second editions, which came out in 1967 for volume 1 and 1969 for volume 2.
The second editions are still the current edition, so no worry that you might be missing out on something if you go with used copies. If you do want new copies (maybe you can't find used copies or they are in bad shape) take a look at international editions.
A new copy of the international edition for India from a seller in India on AbeBooks is around $15 per volume plus around $19 shipping to the US. Same contents as the US edition but paperback instead of hardback, smaller pages, and rougher paper. (International editions also often replace color with grayscale but that's not relevant in this case because Apostol does not use color)).
You can also find US sellers on AbeBooks that has imported an international edition. That will be around $34 but usually with free shipping.
Indian editions sometimes have different question sets to prevent students from using them in other countries' coursework.
They also have a hologram sticker alongside a printed warning that they are not for sale or export outside of India, Nepal and a couple of other countries.
i bought a compilers book that was an Indian edition. The paper and print quality was so bad (like smudgy) that I could not read it and I didn’t think I was particularly picky about this. Not sure if I just got unlucky or if this is generally true?
