# 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?

My book Calculus from the Ground Up focuses on differentials, and uses it to provide a unification of process and simplification of understanding of a lot of different parts of calculus.

To read about the thought process that led to the book you can see this arXiv link; the focus on differentials that you are asking for led naturally to a refactoring of the way introductory calculus is presented.

Differences from other books:

The arXiv link gives some important information, but I'll repeat some of it here. First of all, the focus of the entire book is on differentials. We do a lot of derivatives, but the focus is always on differentials, and for several important reasons. First, it unifies several important practices into a single system - single-variable, multivariable, and implicit differentiation all has the exact same process. Second, it makes the different geometric integrals more obvious. The integral is presented as a sum of infinitesimals, not as an area under the curve (which becomes merely one of the application areas). The integral simply sums up whichever geometry is being used. $$\int y\,dx$$ for summing areas of rectangles, $$\int \pi y^2\,dx$$ for summing volumes of cylinders, and $$\int \sqrt{dx^2 + dy^2}$$ for summing arc lengths. The way you are asked to memorize them is exactly what the geometry states. For instance, many books want you to have the volume of cylinders as $$\pi \int y^2\,dx$$. That's correct, but moving the $$\pi$$ outside means that it no longer looks like the volume of a cylinder equation for students.

Additionally, the book includes a rule that seems to have gone missing for doing differentials of the form $$u^v$$. For those who don't know (because it is missing in most modern books), $$d(u^v) = vu^{v - 1}du + \ln(u)u^vdv$$ (I wish the font for $$v$$ had a more distinct look here, but oh well). Many books teach "logarithmic differentiation" for this, but it is wholly unnecessary. Just like all the other differentials, all you need is the rule.

I also try to include additional life lessons that we can learn from calculus. For instance, in the discussion of Taylor polynomials, I discuss how this can be used as a template for solving impossible problems (not only in math but anywhere).

Also, I wanted to make a note on the second differential, because it came up in the discussion of Keisler. I don't make a big deal about it (I put it in the Appendix), but I actually introduce a form for the second derivative that makes the chain rule for the second derivative work algebraically. Generally, in the text, I avoid this situation by simply introducing a variable for the first derivative, and then take the derivative of that variable. However, in the appendix I show that second differentials can be made algebraic by making the second derivative $$\frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2}$$. If that looks strange to you, you can derive it for yourself by simply taking the derivative of $$\frac{dy}{dx}$$. Note that $$\frac{dy}{dx}$$ is a quotient, so you would use the quotient rule to take the derivative of it. This leads to differentials that are 100% algebraically manipulable. Most texts focusing on differentials don't tell you either the problem nor the solution for using second differentials.

The structure of the book differs from "Calculus Made Easy" in that it starts with derivatives, since a slope is more intuitive for people coming from algebra. Unlike Keisler, it saves discussion of limits for the end of the book. Essentially, it gives you the intuition and the toolset first, and then, at the end, goes into a bit more formally the underpinnings of what makes it work. I find that students prefer this approach. Like Keisler, I use the hyperreal numbers (though I don't formally introduce them until the last third of the book, which focuses on the infinite).

Anyway, I always try to write things in such a way as to focus the student on the intuitions behind everything, so that learning calculus doesn't just teach them calculus, but it improves their thinking. For instance, when talking about the other geometric uses of the integral (volumes from cylinders, volumes from shells, arc lengths, etc.), I gave a general mental mechanism that is used to generate all of these. (a) the problem can be estimated by a formula, (b) the problem can be divided into subproblems, (c) each subproblem must have the same form as (a), (d) the result must be attainable by adding the results of the subproblems, and (e) increasing the number of subdivisions improves the accuracy of the estimation method. The goal here is to show the students how the thought process works.

Also, my student's also love the fact that I show where all of their formulas that they learned in previous math classes come from. I show how to derive the interest rate formula, the volume of a cone formula, and the volume of a sphere formula. In fact, that's another aspect of the book - I teach how to derive formulas. We use calculus to derive the vertex formula for quadratics, and a homework problem is deriving the vertex formula for cubics. I tell students that calculus is the "where babies come from" of math.

NOTE - I edited this to include more details about the book and what makes it different because I was requested to below. Sorry if this comes off as more of an advertisement than was intended.

• I am indeed the author. I'm not sure how it is spam, if the person is literally asking for book recommendations that exactly match the book I am suggesting. – johnnyb Jan 27 '20 at 1:07
• @johnnyb I made an edit to the answer -- does this seem accurate to you? It puts some words in your mouth but I believe it makes it pretty clear that your answer isn't some kind of low-effort spam post. – Chris Cunningham Jan 27 '20 at 20:57
• (Obviously feel free to rework or revert what I wrote, but mentioning that you are the author is just good practice.) – Chris Cunningham Jan 27 '20 at 20:59
• Thanks for the help, Chris! – johnnyb Jan 28 '20 at 16:32
• The links goes straight to Amazon, where the "author, teacher, researcher and computer programmer" offers a book upending "the general method of calculus training" that have been "set in stone for the last hundred years," but offers no preview. There are many books that leave out limits for later - or never introduce them - like a century-old classic by Silvanus Thompson, which can be accessed free online and which is mentioned by the author. It would be lovely if the author briefly explained the differences and benefits of his book compared to similar books, not to dissimilar ones. – Rusty Core Jan 28 '20 at 22:29

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