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Consider the following example of working "directly" with differentials.

One way to approach the problem of determining the arc length of the graph of a single-variable function is to imagine the arc broken up into infinitesimal line segments. Such a line segment is the hypotenuse of a right triangle with legs of length $dx$ and $dy$ (infinitesimal changes in $x$ and $y$, respectively). By the Pythagorean Theorem, the length of the hypotenuse is $$ \sqrt{(dx)^2 + (dy)^2}, $$ which one can rearrange to either $$ \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \ dx $$ or $$ \sqrt{\left( \frac{dx}{dy} \right)^2 + 1} \ dy $$ depending on whether it is easier to integrate with respect to $x$ or $y$, respectively.

In general, can one be led astray by reasoning directly with differentials in this way (as opposed to defining a Riemann sum and taking a limit)? If so, are such examples difficult to contrive or are they typical enough that arguments like the one I presented should be avoided?


Edit: I feel my question was not clearly stated. By "can one be led astray", I mean I am concerned about other applications in Calculus or Physics where a perfectly sensible-looking manipulation of differentials (like the one in my example) leads one to an incorrect conclusion.

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    $\begingroup$ I teach it this way. $\endgroup$ – Sue VanHattum Feb 10 '15 at 13:56
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    $\begingroup$ Better yet, one can prove the derivative of sin(x) is cos(x) using differentials like this instead of having to compute $\lim_{h \to 0} \sin(h)/h$. $\endgroup$ – Aeryk Feb 10 '15 at 22:31
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    $\begingroup$ Perhaps what you want is to make a standard disclaimer for most of what you do in your class: The way we're going to study calculus is the way everyone does it. We will pay little mind to what "dx" and "dy" mean precisely, but only give them loose interpretations as "very small" numbers. The historical development of calculus is littered with intense debates as to what "dx" and "dy" actually mean and the nature of the idea of a limit in general. These debates took roughly 200 years to resolve, and so if you feel uncomfortable following any part of our reasoning, you would have been in good $\endgroup$ – Tac-Tics Apr 21 '17 at 2:31
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    $\begingroup$ (cont.) company. If you are truly curious, though, about the rules that govern the subject, the name for the rigorous study of calculus goes by the name of analysis. But while calculus was developed during the 1600s, it wasn't until the 1800s that the foundations were finally established. If our reasoning was good enough for Newton and Leibniz, it will be good enough for our class. [repost from a converted answer] $\endgroup$ – quid Apr 22 '17 at 19:52
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As @BenCrowell mentioned, the transfer principle proves that direct algebraic manipulation of infinitesimals in single variable calculus is allowed.

But I stumbled upon this post in theshapeofmath.com, which shows how things can break when switching to multiple variables. It provides the following basic example of a possible error when handling partial derivatives (the same example also appears as an answer in the MO question linked to by @AmirAsghari), where given an implicitly defined function $F(x,y)=0$ , we can algebraically get an expression with the wrong sign: $$\frac{dy}{dx}=\frac{\partial F}{\partial x}\frac{\partial y}{\partial F}=\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}.$$

While the correct expression is: $$\frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}.$$

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    $\begingroup$ NSA and the transfer principle aren't limited to single-variable calculus. The example you give that results in an incorrect answer is a good one, but the mistake is a mistake in logic, not a mistake resulting from the use of infinitesimals. $\endgroup$ – Ben Crowell Feb 16 '15 at 19:14
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In general, can one be led astray by reasoning directly with differentials in this way (as opposed to defining a Riemann sum and taking a limit)? If so, are such examples difficult to contrive or are they typical enough that arguments like the one I presented should be avoided?

People used differentials for centuries before the calculus was reformulated in terms of limits. Nothing bad happened as a result of these procedures, and they can be more formally justified in non-standard analysis (NSA). NSA has a rule called the transfer principle, which guarantees that any of the elementary axioms of the real number system can be applied to differentials as well. ("Elemenary" means anything except for the completeness property.) Since the manipulations in your question can all be justfied from the elementary axioms, they are all OK.

The main thing you have to watch out for is identifying dy/dx with the derivative y', or the closely related practice of discarding squares of differentials. The more rigorous way to do this is to talk about the standard part of a number, which means the unique real number that differs from it only infinitesimally. Then the derivative is defined as the standard part of dy/dx. Leibniz actually made this distinction, but other people got sloppy later on and dropped it.

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One can be misled by anything and everything. However mathematicians have written things like $$ ds^2 = E\, du^2 + 2F\, du\, dv + G\, dv^2 $$ for a long time without problems.

LINK

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  • $\begingroup$ I don't think it's accurate to say that this is "without problems". I don't have a particular example for confusion caused by this statement, but I've heard of many cases where abuse of notation caused false analogies, both in student learning, and in professional proofs. It's not incidental that rigor is so highly valued in mathematics. $\endgroup$ – yoniLavi Feb 13 '15 at 21:30
  • $\begingroup$ @yoniLavi: But this is not an abuse of notation, nor does it lack rigor. $\endgroup$ – Ben Crowell Feb 15 '15 at 2:21
  • $\begingroup$ @BenCrowell, I won't argue about this specific example, since I don't know enough about it. What I meant is that in general algebraic manipulation of differential forms, such as this, is not trivial and needs to rely on rigorous rules derived either from non-standard analysis or epsilon-delta proofs, both of which were developed long after such statements had been originally written by Leibnitz. $\endgroup$ – yoniLavi Feb 15 '15 at 5:42
  • $\begingroup$ @yoniLavi: Nobody said anything about differential forms, and differential forms aren't the same thing as the infinitesimals of NSA. algebraic manipulation [of infinitesimals] is not trivial The transfer principle basically says that it is trivial. $\endgroup$ – Ben Crowell Feb 16 '15 at 2:44
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    $\begingroup$ @yoniLavi: There are many different ways of viewing this sort of thing. Differentials forms are one way. You may also be interested in the following: mathoverflow.net/questions/186851/… arxiv.org/abs/1405.0984 $\endgroup$ – Ben Crowell Feb 16 '15 at 19:11
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Since the rearranging part of your strategy involves dividing $dy$ by $dx$ this post on MO is quite related to your concern.

As far as the arc length is concerned, I usually avoid your strategy as a teaching tool at the outset, but sometimes mention it as a way to memorize the formula. Why do I avoid this? I have no conceptual way to distinguish between $dS$ (the infinitesimal arc length) and $dx$. They are infinitesimally small so "intuitively" they have to have the same length; in particular $dy$ is also infinitesimally small give additional support to the intuition: In a right triangle with such an small side $dx$ is equal to the hypotenuse ($dS$). Right? Indeed, once as a student, I did something similar to this line of argument when calculating the area of a surface of revolution for the first time on my own.

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It has long been a puzzle to historians how Leibniz could not have been misled by using differentials, given that to him "differential" meant "infinitesimal" and infinitesimals were thought to be inconsistent. About 50 years ago Abraham Robinson clarified the picture by providing a justification for infinitesimals that's rigorous by current standards. There is little room today to think that a justification by means of differentials or infinitesimals should be more prone to error than any other kind of justification.

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It's easy to confuse yourself with differentials. Consider the surface area of a cone given in cylindrical coordinates by $z=1-2r$. What is the element of area?

Should I visualize a cylinder tangent to the cone? \begin{equation} \begin{aligned} dA &= (d\text{ curved base})(d\text{ height}) \\ &=(r\,d\theta)\sqrt{dr^2+(2\, dr)^2} \\ &=\sqrt{5}\, r\, dr\, d\theta \end{aligned} \end{equation} Or should I visualize a rectangle tangent to the cone? \begin{equation} \begin{aligned} dA &= (d\text{ linear base})(d\text{ height}) \\ &=r\sin(d\theta)\sqrt{dr^2+(2\, dr)^2} \\ &=\sqrt{5}\, r\,dr\sin(d\theta) \\ &<\sqrt{5}\, r\,dr\,d\theta\, (?) \end{aligned} \end{equation} I relied on the formulas in Thomas & Finney for years before I could confidently use differentials in a derivation like this.

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    $\begingroup$ If $\sin(d\theta)$ has any meaning at all, it should be equal to $d\theta$, since $\sin(d\theta) = d\theta - (d\theta)^3/3!+... = d\theta$, as higher powers of $d\theta$ vanish. So I do not see this computation as "leading one astray". $\endgroup$ – Steven Gubkin Feb 16 '15 at 20:01
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    $\begingroup$ @StevenGubkin, I agree on the meaning of $\sin(d\theta)$, but that manipulation with the power series was not obvious to me when I was first learning how to compute surface area. Just $(d\theta)^3=0$ on its own would lead some students astray -- at a minimum, differentials like this can easily lead students astray from the path of confident understanding. $\endgroup$ – user173 Feb 16 '15 at 22:10
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Before I answer I'd like to clarify that I've never taught this topic and that any teaching should be adapted to the learners' needs. That said the topic itself suggests we are talking about talented students.

An important question for me is "What are the students learning?".

I would argue they are practicing manipulating variables. In particular they are realizing they can put variables in and out of root signs by either squaring or square rooting.

I would also say they are becoming familiar with the arc length formula and calculating it.

I don't think there is anything wrong with these learning outcomes at all.

A different question is "What could they be learning?"

I think there is an opportunity for them to be learning about the importance of approximations. This to me is a deeper and more challenging skill. I struggle myself if I'm honest. I'm not necessarily suggesting they learn the full rigours of a Riemann sum but perhaps a halfway house? I reiterate I have no experience at this level. However asking students to approximate the length of a line with fixed $\delta x=0.2$ would be a useful exercise. What happens if $\delta x =0.1$? I feel that the limiting process involved in integration is too readily forgot not least by myself. Approximating other objects or circumstances and forming other integrals might be less of a daunting prospect for some once they familiarize themselves with easier examples.

To summarize I think working with infinitesimal fine but I don't think I'd reason with them. Hope this helps.

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I think there is nothing wrong with reasoning using differentials/infinitesimals in introductory calculus and physics classes provided the nature of the reasoning is made clear and explicit. More precisely, my contention is that in general such uses serve to motivate definitions rather than to hide hard-to-formalize details of proofs, and it is operationally irrelevant whether such motivational uses are formally justifiable.

(I am conflating differential/infinitesimal here because, although in general I would not conflate these notions, in the context of the question that was asked, it seems to me that a differential $dx$ is being treated as something very small but not zero by which it makes sense to divide, so the while the notation is for differentials the sense is for infinitesimal.)

Personally, when I teach these things, I do not write $dx/dy$ to mean divide the very small $dx$ by the very small $dy$. I write instead $\Delta x$ and $\Delta y$ and $\Delta x/\Delta y$ which I speak of as change in $x$ and change in $y$ and treat as small, but nonzero, quantities. I do this because I don't want students to get in the habit of operating too freely with the Leibniz notation, not because I don't like the notation, but because experience suggests that free manipulation of such notation causes a lot of difficulties for students inclined to interpret everything as a series of formal manipulations.

In the specific example of the arc length of a curve, it is important to keep in mind that without calculus one has no definition of the arclength of a general curve. One knows what one means by the length of a line segment (having fixed beforehand a length scale), but extending this even to graphs of polynomials is not obvious (it is exactly the sort of problem that gave rise to calculus).

The naive approach, which turns out to work, is to approximate a given continuous curve (but what does continuous mean without some notion of limit/infinitesimal? - so if you like restrict to the graphs of polynomials or real analytic functions or some such class of curves whose members are de facto continuous but where this does not have to be proved) by a polygonal curve joining some points along the curve that partition the curve into subcurves. It is natural to view the length of this polygonal curve as an approximation to the as of yet undefined length of the curve in question. Doing so one obtains a sum plus some errors terms, and the sum in question can be interpreted as a Riemann sum approximating some so far unformalized integral. Moreover, its form does not depend on the chosen partition (of course its value does). The error terms can be controlled formally using the Taylor approximation, or can be treated as infinitesimals in some hand wavy way - it really doesn't matter - because one is not going to prove that in the limit the errors are negligible and the remaining sum converges. Rather, one supposes the limit exists, or, more properly, calls the curve rectifiable provided the limit of the sums exists, in which case that limit is precisely the integral for which the sums were viewed as approximations. Hence one says the curve is rectifiable if the corresponding integral exists (its precise form is irrelevant to this discussion), and when it does calls its value the length of the curve.

That this is a reasonable definition is not something to be proved beyond the almost trivial observation that applied to the primitive objects used to motivated the definition (that is to line segments) it recovers the primitive notion of length, and in some other essentially physical examples, such as the circumference of a circle, that it yields the known (empirically) answer. Consequently, it matters not at all that some informal notion of infinitesimals was used in motivating the definition. Its correctness (beyond its self-consistency) is a thing judged by empirical and aesthetic standards external to the mathematics.

In general, most uses of infinitesimals in introductory calculus and physics courses follow this paradigm. The purpose of their use is not to provide an informal or heuristic verification of something that could otherwise be proved. Rather, it is to motivate the extension of some primitive concept, such as length, area, work, circulation, etc. to a class of geometric objects for which the primitive notion is by itself inadequate. The correctness of this extension is judged by its utility or mathematical beauty, or whatever other external criteria are appropriate.

One can go astray in this sort of reasoning, and this does happen when it is applied in more complicated physical contexts where there may be more than one natural way to realize the approximation of whatever thing needs to be computed (defined! to compute something one first has to define what it is one computes) and they may lead to different limiting interpretations, with different results. In physical contexts one is correct, one is not, but this is judged comparison with experiment (something like this underlies the discrepancy between the vakonomic versus nonholonomic interpretations of velocity dependent constraints in classical mechanics - one of which is generally incorrect for real physical systems). In purely mathematical contexts one might (if one gets lucky!) obtain two different theories, and, while one might be judged more interesting than the other, it might be hard to call one more correct than the other.

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