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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?

Michael Hardy
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Mike Shulman
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7 Answers7

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This might be taking things too far, but Keisler'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.

kcrisman
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Kevin O'Bryant
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    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
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    On general principle, I always upvote comments about general principles. So we're good. – Kevin O'Bryant Mar 16 '14 at 00:01
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    @MikeShulman: Interesting! What is it about Keisler's definition of second differentials that you find incorrect? – String May 12 '14 at 08:53
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    @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
  • @MikeShulman: Ah OK. Another way to recover it is to assume $d^2 x\equiv 0$. But then mixing second order derivatives with respect to different independent variables will have issues as we discussed in a thread in Mathematics Stack Exchange. – String May 12 '14 at 20:28
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    @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
  • I have reviewed Keisler's definition of second differentials via infinitesimals and found it to be correct, just like his definition of first differentials. – Mikhail Katz Feb 11 '16 at 13:47
  • The problem is that his definition of second differentials yields the wrong chain rule for second derivatives. Of course a definition can't technically be "wrong", but since the fact that you can get the chain rule for first derivatives by simply substituting into differentials is one of the main advantages of differentials (in my opinion), having this fail for a definition of second differentials means that that definition is not very good. – Mike Shulman Feb 11 '16 at 17:05
  • @Mike, Leibniz and the rest of the pack talked about "constant differentials" when they wanted to force $d^2 x=0$. In modern terms this corresponds to having a hyperfinite partition into equal infinitesimal segments. Of course if $dx$ are "constant" this does not mean that the $du$ are constant. So if $x$ is the independent variable it seems reasonable to assume that the $dx$'s are constant, and then we have the universally accepted formula $f''(x)=\frac{d^2 y}{dx^2}$, which isn't any different from what Keisler is saying. So I really don't understand what you are objecting to here. – Mikhail Katz Apr 25 '17 at 08:53
  • @MikhailKatz For first differentials, if $u=f(x)$ and $y=g(u)$, then $du = f'(x) dx$ and $dy = g'(u) du$, and by substituting $u$ and $du$ into this expression for $dy$ we get $dy = g'(f(x)) f'(x) dx$, which gives the correct chain rule formula $dy/dx = g'(f(x)) f'(x)$. But if we write $d^2u = f''(x) dx^2$ and $d^2y = g''(u) du^2$, then by substituting $u$ and $du$ into this expression for $d^2y$ we get $d^2y = g''(f(x)) (f'(x))^2 dx^2$, which would give the incorrect formula $d^2y/dx^2 = g''(f(x)) (f'(x))^2$ for the second derivative. – Mike Shulman Apr 25 '17 at 11:07
  • The correct chain rule for the second derivative is obtained by writing instead $d^2u = f''(x) dx^2 + f'(x) d^2x$ and $d^2y = g''(u) du^2 + g'(u) d^2u$, whence $d^2y = (g''(f(x)) (f'(x))^2 + g'(f(x)) f''(x)) dx^2 + f'(x) d^2x$, in which the coefficient of $dx^2$ is the correct second derivative $(g\circ f)''(x)$. From this perspective it is more correct to write $\partial^2f/\partial x^2$ than $d^2f/dx^2$ for the second derivative in general. – Mike Shulman Apr 25 '17 at 11:08
  • @Mike, I agree with the mathematics :-) In other words, you can't consider both $x$ and $u$ as having "constant differentials" unless they are proportional. But Keisler's book uses extensively the notions of dependent variable and independent variable. If you assume that $x$ is the independent variable, it cannot be said that Keisler's definition is incorrect. What you seem to be saying is that no variable should be assumed independent and every variable should be held to be potentially dependent. Does any textbook follow this practice? – Mikhail Katz Apr 25 '17 at 11:13
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    @MikhailKatz I'm not saying anything philosophical about independent or dependent variables or "constant differentials". What I'm saying is that one of the main computational advantages of differentials is that you can get the chain rule by simple substitution, and that that advantage should be maintained in any definition of "second differential". Whatever that means in terms of "independent and dependent variables" (which I don't really consider to be a mathematical notion). – Mike Shulman Apr 25 '17 at 11:31
  • @Mike, Keisler is more of a mathematical logician than I am as you know, and he considers the distinction to be sufficiently mathematical to claim that $dx=\Delta x$ for the independent variable whereas $dy\not=\Delta y$ in general for the dependent variable. I see nothing wrong with that. – Mikhail Katz Apr 25 '17 at 11:34
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    @MikhailKatz Of course, when you take a derivative, you have to fix one variable to be the "input" and another to be the "output" (value of the function you are differentiating). When taking partial derivatives it's even worse: you also have to fix the other independent variables that you're not differentiating with respect to. But one of the advantages of differentials is that we don't have to worry about making such choices; any relationship between variables induces a differential relationship. – Mike Shulman Apr 25 '17 at 14:58
  • Maybe this is a subject for a separate question but what I am interested in, in the context of the second derivative, is to be able to write it as a second difference divided by $h^2$, where the second difference is something like $f(x)-2f(x+h)+f(x+2h)$. I wonder how this is affected by your preferred approach. – Mikhail Katz Apr 25 '17 at 15:07
  • @MikeShulman... – Mikhail Katz Apr 25 '17 at 16:20
  • Let us continue this discussion in chat. – Mike Shulman Apr 25 '17 at 19:14
  • Infinitesimals don't really contribute to the definition of differential at all; Keisler is using a standard definition of differential, which doesn't depend on $\Delta x$ being restricted to infinitesimal values. –  May 03 '17 at 12:19
  • @Hurkyl if you use infinitesimals, though, the error is also infinitesimal. – Christopher King Feb 09 '19 at 01:39
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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.

Community
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Michael Bächtold
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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.

johnnyb
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    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 01:07
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    @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
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    (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
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    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
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    @RustyCore - I added some details for you! Also note that, at the moment, while there is a Kindle edition, for some reason it is not linked to the main book, but can be found by searching. A solution guide is also available, which gives detailed workings of each problem in the book. – johnnyb Jan 29 '20 at 13:00
  • Any way making the book available for Kindle Unlimited program? I skimmed through the intro of the book available on Amazon. In the intro you say that the equation of non-uniform motion should be memorized unless one knows calculus, in which case it can be easily derived. Well, in the case of constant acceleration (this is how you wrote it) it can be easily derived without calculus, so calculus may seem like a hammer that can be used for any nail, large and small. – Rusty Core Jan 30 '20 at 21:41
  • "Derivatives are introduced through extensive practical work on evaluating slopes" - if physics was a standard course in American schools, you would be calculating speed and distance of non-uniform motion, introducing derivative and integral in one simple real-life example. In fact, you can still do that without referring to a physics course. Abstract slopes are no more tangible to students than limits. – Rusty Core Jan 30 '20 at 21:42
  • In the intro you say that the book is written in a conversational style. In your estimation, has this affected the size of the book? It is 400+ pages long. Could it be condensed into, say, 200 pages, but with more concise language? Not everyone enjoys reading textbooks written like novels, especially when a test is tomorrow (it is always tomorrow). "The book is written in a way I would talk in my class" - Strang's book, written in a way he talks in his class, is an incoherent stream of conscience. The Thompson's is also written in a relatively relaxed style, but feels quite tight. – Rusty Core Jan 30 '20 at 21:46
  • As far as Kindle unlimited, it did not give me that option (probably because of the price). If you are looking to use this somewhere, I'm happy to send you a physical copy. As for your questions:
    1. it can be easily derived without calculus - there's a lot of things can be derived in multiple ways. The fact that there is more than one way to do it is not very relevant.
    2. "Abstract slopes are no more tangible to students than limits". Not really. Students coming into calculus spent a lot of time with slopes when studying lines. This is already something they are familiar with.
     – johnnyb Feb 02 '20 at 21:24
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  • "conversational style. In your estimation, has this affected the size of the book?" Absolutely. I mean, if you want it really concise, I've got an appendix that has Algebra in six pages and Calculus in three. The point of the book is to teach students. It's written to learn from. I think you would have to read it to know if it is a style you like, but I get a lot of positive feedback from my students. Also, when the test is tomorrow, the number of pages doesn't matter, only the understandability of those pages. I've had 3 pages that took longer to read than 20 that were explained well.
    • – johnnyb Feb 02 '20 at 21:28
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    I am accepting this answer because whatever its other merits or flaws may be, this book does, I believe, come the closest to the sort of thing I was looking for when I asked the question. Especially the honest treatment of the second derivative! – Mike Shulman Mar 13 '20 at 23:27