# Calculus book suggestion

I am a high schooler with no prior exposure to calculus. I want a calculus book to learn math for classical mechanics on my own, and perhaps learning math for math itself. I don't like memorizing formulas, I want some understanding but nothing too rigorous.

A lot of people suggest Thomas and Stewart, but a lot people dislike them as well. Why do people dislike these books? Because they are too simplistic? And they came overly long to me (over 1000 pages).

I think Lang, and perhaps Kline are nice, but I am not sure. And it came to me that these books are better in 'why's of formulas. And there is Simmons as well.

I worked through Spivak a little, but it was too hard. Perhaps I may return to it after some exposure to calculus to refine my understanding of the concepts.

Thanks for suggestions.

• Not a book, but as for building intuition, 3blue1brown has an excellent video series called “essence of calculus” which gives visual and conceptual intuition for the subject – eepperly16 Jan 17 '18 at 17:54

Thomas and Stewart are both standard texts used in introductory calculus classes. They are of a kind—they are not terribly rigorous, break things down into tiny little digestible pieces, but neatly avoid most of the interesting theory. If your only goal is to learn enough calculus to perform computations, either book would be fine. That being said, I would recommend against them—I wouldn't consider their use "unethical," but I don't think that they are good introductions to what mathematics really is.

I might suggest one of the following:

• Lang, Serge, A first course in calculus. 5th ed, Undergraduate Texts in Mathematics. New York etc.: Springer-Verlag. XV, 624, A99, I3 p. DM 128.00 (1986). ZBL0608.26001.

Lang is pretty approachable, and has some nice examples. Lang's text also seems to find a nice middle-ground between the rigour and application—he elides a lot of the theory (though not to the point of uselessness), and does a good job of giving intuition. It is also possible that you might be able to find an older, cheaper edition running around somewhere. Lastly, my recollection is that there is a really nice section on Stirling's formula near the end, which may be worth the price of admission.

• Courant, Richard, Differential and integral calculus. Vol. 1. New edition, Glasgow: Blackie & Son. xiv, 615 p. (1937). ZBL0018.30001,

There are actually two volumes here. Contrary to many modern authors, Courant first introduces the integral, then differentiation. Personally, I think that this is a lovely and natural approach, though your milage may vary. The writing is a bit old-fashioned, but this may be good thing. ;)

• Apostol, Tom M., Calculus. Vol. I: One variable calculus, with an introduction to linear algebra. 2nd ed, A Blaisdell Book in the Pure and Applied Sciences. Waltham, Massachusetts-Toronto-London: Blaisdell Publishing Company. A division fo Ginn and Company. xx, 666 p. (1967). ZBL0148.28201.

I am less familiar with this particular text (again, I think that there are two volumes), but it comes highly recommended from colleagues. In other texts, Apostol is quite approachable—his exposition is generally quite good, and he builds up abstract ideas in a clean manner.

• Klein, Morris., Calculus: An Intuitive and Physical Approach. 2nd ed. New York, John Wiley & Sons, Inc. (1977).

I recently picked up a copy of this text, so am adding it to the list since I think that it is very nice. The exposition is clear (though perhaps a bit verbose), it is rigorous, and does a good job of going over the standard curriculum. It also includes some good historical context, and spells out notational conventions which often trip up beginners (for example, his discussion of $$\Delta t$$ as a single symbol is spot on).

If you hadn't already ruled it out, I might also have suggested Spivak, but, as you say, his text requires a bit of work to get through. The work pays off, but Spivak can be intimidating, and it is understandable that you might want to consider something a bit more approachable.

Stewart is the standard. I happen to like it a lot. Also, if you are working through it on your own the nice part about it is you can purchase the accompanying student solutions manual that has all the exercises worked out for you. The overly long edition is likely the one that includes multivariable calculus. There is a shorter edition that does not include it. With regards to the lengths of these books in general, unfortunately there are no shortcuts in learning, the only way to learn the material is to...learn the material :(

• Stewart is utterly conventional, and not bad for some students. Today students who work hard and want to get good grades but have no desire to understand calculus are most of those who show up in the kind of course for which it is used, and it's unsuitable, to the point of being unethical, to use it for those. – Michael Hardy Jan 17 '18 at 18:36
• @MichaelHardy Hey Michael! Thank you for your thoughts. I happen to like it. I said it was the standard because it was my text for Cal in both high school and college, and my sister used it for hers at a totally separate college. I think understanding calculus is more for a real analysis class. Introductory calculus is more to teach ppl how to "use" calculus, as that's all a lot of professions need. I encourage you to add your own recommendations on this page that you think may be a better choice. – David Reed Jan 17 '18 at 18:51
• Stewart's textbook is nowhere near being a text whose emphasis is on using calculus. – Michael Hardy Jan 17 '18 at 19:16
• A book on using calculus probably wouldn't start by going through all that stuff about limits at the beginning; it probably wouldn't try to cover every way to compute derivatives by elementary means; it probably wouldn't give justifications of all of its methods of computing derivatives; and its section purporting to give applications probably wouldn't be dishonest. – Michael Hardy Jan 17 '18 at 19:23

Calculus by Thomas Finney has a nice modernized approach to it with computer based problems and hyperlinks to various visualization and images .

• I second this suggestion – XRBtoTheMOON Jan 17 '18 at 18:31

Xander's answer is good. It's true that it depends on your goal. Elementary caluclus classes are not a good representation of what mathematics is. They tell you the rules which you should follow without investing in the necessary knowledge for you to understand why the rules are the way which they are.

I found this book truly illuminating: Lay. It is very popular and frequently used in classes at the university level for math majors which intend to go back and correct the sloppy job done by Calc 1-4. This book is typically used in classes which require calculus as a prerequisite. However, I believe that is a mistake. You can absolutely read this book with zero pre-requisite knowledge, so long as you pace yourself and come prepared to grapple with a few abstract concepts.

If you just want to have calculus as a tool without really understanding why it works (which is fine, depending on your career goals and generally philosophy), then Stewart will do. However, it is both pedagogically and logically more correct to invest in the prerequisite knowledge in order to truly understand a bit of calculus.