# A calculus book that uses differentials?

All introductory calculus books that I have seen spend most of their chapters on differential calculus talking about derivatives, with at most a short section defining differentials as $dy = f'(x) dx$. However, differentials are useful for understanding a lot of things, like linear approximation, the chain rule, integration by substitution, and (when you get to multivariable calculus) the change-of-variables formula and the various manifestations of Stokes' theorem. One doesn't have to agree with everything that Dray and Manogue say to want to try introducing and emphasizing differentials early in differential calculus.

Is there any calculus textbook which does such a thing?

This might be taking things too far, but Kiesler's book (available free online) does everything using infinitesimals, which make differentials literally immediate. The rigorous underpinning for infinitesimals is nonstandard analysis, but this book doesn't dwell on that. It just teaches how to use them correctly.

I'm guessing this isn't exactly what you were looking for, but it might be worth checking out because it's free.

• Thanks! I've looked at Keisler's book before, and considered it seriously. In general, I think infinitesimals are actually orthogonal to differentials: one can use either one without the other. However, Keisler does use differentials fairly seriously as well (although he defines the second and higher differentials incorrectly in my opinion), so this would be worth an upvote. Unfortunately, on general principle I never upvote an answer that explicitly asks to be upvoted. (-: – Mike Shulman Mar 14 '14 at 21:50
• On general principle, I always upvote comments about general principles. So we're good. – Kevin O'Bryant Mar 16 '14 at 0:01
• @MikeShulman: Interesting! What is it about Keisler's definition of second differentials that you find incorrect? – String May 12 '14 at 8:53
• @String One of the important aspects of differentials, especially for a calc 1 class, is "Cauchy's invariant rule": that you can do the chain rule by substitution. That fails for second derivatives using Keisler's definition $d^2f=f''(x)dx^2$. To recover it you need instead $d^2f=f''(x)dx^2+f'(x)d^2x$. I learned this from Toby Bartels. – Mike Shulman May 12 '14 at 20:09
• @String I'm not sure what you mean by "recover". Of course if you assume $d^2x=0$ then $d^2f=f′′(x)dx^2+f′(x)d^2x$ reduces to $d^2f=f′′(x)dx^2$, but the point is that the latter formula gives you the wrong chain rule. E.g. if $y=f(u)$ and $u=g(x)$ then from $d^2y=f''(u)du^2$ and $d^2u=g''(x)dx^2$ you get by substitution $d^2y = f''(g(x)) (g'(x))^2 dx^2$ which is not the correct second derivative of $y = f(g(x))$. – Mike Shulman May 13 '14 at 21:13

Was Silvanus Thompsons lovely "Calculus made easy" mentioned already? It's a classic (100 years old) freely available on gutenberg.com. Some opinions of it can be found on mathoverflow.

It doesn't go very far so it might need to be supplemented with another text, but I believe it does a great job at teaching the physical and geometrical intuition on differentials. It seems that it's closer to synthetic differential calculus than to non-standard analysis in the way it treats infinitesimals.

• It's a very nice book, but one can't really use it as the primary textbook for a class, can one? – Mike Shulman Jul 4 '14 at 16:06
• Good question. I haven't tried. The good thing is that one may modify the original book to fit ones purposes according to the gutenberg license – Michael Bächtold Jul 4 '14 at 18:31