Is it a dead end to define differentials as finite differences on the tangent line?

Question

In a freshman calculus text (Larson), I was surprised to find a definition of differentials as finite differences on the tangent line, and even more surprised to learn later that this definition apparently dates back to Leibniz. My main source on the history is Boyer, The History of the Calculus and its Conceptual Development, 1939, which is available online. Boyer characterizes Leibniz's definition as follows (p. 210):

In the first published account of the calculus, Leibniz gave a singularly satisfactory definition of his first-order differentials. He said that the differential dx of the abscissa x is an arbitrary quantity, and that the differential dy of the ordinate y is defined as the quantity which is to dx as the ratio of the ordinate to the subtangent.

Boyer then goes on to describe the fact that this approach didn't work for higher-order differentials, so that Leibniz ended up using a mixture of the above definition and a framework involving infinitesimals. In Larson's language, dx is "any nonzero real number."

Yesterday I interrogated a few students who had learned calc from the Larson text. I asked them something like this: You do integrals using substitutions, and during that process, you end up writing things like du=2dx. What kind of things are du and dx? I got blank stares, so I asked if they were numbers, small numbers, ...? The reaction I got was basically that they had learned a technique of integration called u-substitution, and that there was a set of procedures for doing that, which entailed manipulating symbols in this way.

Is Larson's approach common pedagogy? In this approach, how does one get past the issues with higher-order differentials -- do these issues make this approach a dead end? Should this be understood as some kind of watered-down presentation of differential forms?

Another thing that seems odd to me is that Larson writes things like $\Delta y\approx dy\approx 0.06$. I think the vast majority of scientists and engineers would consider this mathematically illiterate or a naive mistake, so it seems like Larson's approach has the disadvantage of being incompatible with the way the rest of the world talks about differentials.

Related:

• Practical experience with teaching differentials in freshman calc?
• A calculus book that uses differentials?

Answer 1

The senior faculty at my institution were also unhappy with the presentation of the "differential" in the style you describe. I gather they were unhappy with the appearance of mixing $dx$ with $\triangle x$ and the lack of notation to indicate the base point of the approximation. That said, it is often the case that instructors slavishly follow the text so I would wager the pedagogy is quite common.

My approach in first semester calculus is simply to emphasize the linearization of the function at a point. In particular, I define $L_f^a(x) = f(a)+f'(a)(x-a)$ and we learn that $y= L_f^a(x)$ is the tangent line. I try to drive home the concept that this gives us the best affine approximation to the function near $x=a$. Or, equivalently, $L_f^a(x)-f(a) = f'(a)(x-a)$ gives us the best approximation to the change in the function near the point. Of course, you can see how this notation is a bit of a drag so it's helpful to invent some simple notation $\triangle x = x-a$ and $\triangle y = f(x)-f(a)$ thus $\triangle y = f'(a)\triangle x$. I see no need for $dx$ or $dy$ here.

Let me discuss the differential notation(s). If the differential of $f: \mathbb{R} \rightarrow \mathbb{R}$ is denoted $df_a:\mathbb{R} \rightarrow \mathbb{R}$ then $df_a(h) = f'(a)h$. In Edward's Advanced Calculus text he distinguishes the differential $df_a$ and the derivative $f'(a)$. This terminology continues for mappings $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ where $dF_a$ is a linear transformation whereas $F'(a)$ is the Jacobian matrix.

It is clear enough that $df_a$ is a linear form on $\mathbb{R}$. However, for the student who goes on to deeper things, if $f: M \rightarrow N$ then $df_a: T_aM \rightarrow T_{f(a)}N$ thus (returning to the case $M=N=\mathbb{R}$) $df_a(h)$ does not even make sense. Unless of course, we identify $\frac{d}{dt}|_a = 1$ hence the vector $h\frac{d}{dt}|_a = h$. Naturally, this is the identification which is made. Here I assume that we take the set of derivations at $p \in M$ as the tangent space to the smooth manifold $M$ at $p$.

Notice, as we transition to discussing differential forms which take tangent vectors in their domain to return real values we typically have an isomorphism or two which are oft used without explicit mention. In my current formulation of advanced calculus I spend about half the semester with traditional column-vector based differentials then the later half of the course uses the more abstract differential forms which eat derivations.

Your comment about Leibniz getting confused about higher derivatives reminded me of my own confusion when I approached a question about higher derivatives of $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$. I don't know if we really appreciate this enough: higher derivatives calculus I style are a luxury. A fortunate accident that we can identify linear forms on $\mathbb{R}$ with real numbers themselves; $\alpha: \mathbb{R} \rightarrow \mathbb{R}$ linear is naturally identified with $b \in \mathbb{R}$ for which $\alpha(x)=bx$. In the same way, we replace $df_a$ with $f'(a)$. These naturally correspond. Moreover, the function defined by $x \rightarrow f'(x)$ is easy enough to differentiate again as to define $f'': \mathbb{R} \rightarrow \mathbb{R}$ in the same point-wise fashion. However, as you think about the deeper definition, the differential is basic. The second derivative must study the change in the map $x \rightarrow df_x$. This is a whole other ball game. Now, the output is not a number, it is a mapping. I'll not try to explain the resolution to this here, but, beware, this is something hidden by our reluctance to properly embrace the centrality of the linearization idea in calculus. But, don't take my word for it. Let me quote a master:

Dieudonne said it best: this is the introduction to his chapter on differentiation in Modern Analysis Chapter VIII.

The subject matter of this Chapter is nothing else but the elementary theorems of Calculus, which however are presented in a way which will probably be new to most students. That presentation, which throughout adheres strictly to our general ''geometric'' outlook on Analysis, aims at keeping as close as possible to the fundamental idea of Calculus, namely the ''local'' approximation of functions by linear functions. In the classical teaching of Calculus, the idea is immediately obscured by the accidental fact that, on a one-dimensional vector space, there is a one-to-one correspondence between linear forms and numbers, and therefore the derivative at a point is defined as a number instead of a linear form. This slavish subservience to the shibboleth of numerical interpretation at any cost becomes much worse when dealing with functions of several variables...

Dieudonne's then spends the next half page continuing this thought with explicit examples of how this custom of our calculus presentation injures the conceptual generalization.

To summarize, using $dx$ and $dy$ for linearization may be customary, but it makes it all the more difficult to cleanly articulate linearization. The focus ought to be more on the best linear approximation concept as opposed to a slick-looking formula.

(to be clear, I'm not advocating Dieudonne for first semester calculus)

Answer 2

Differentials are not infinitesmals!

Larson's definition is not a watered down presentation of differential forms: it is exactly an explanation of what $df$ is as a differential form when $f:\mathbb{R} \to \mathbb{R}$. In particular $df\big|_p(h) = f'(p)h$. We have the fundamental approximation that $f(p+h) \approx f(p)+df\big|_p(h)$. I suppose you could rewrite this approximation as $\Delta f \approx df(\Delta x)$, if you understand that this applies at each basepoint, and that $\Delta f$ is the change in $f$ resulting from incrementing $x$ by $\Delta x$. So it would probably be better to write something like $\Delta y \approx dy(\Delta x) \approx 0.6$.

Answer 3

Consider the notion of the "tangent space" to a differential manifold at a point.

There are various ways to make this abstract notion concrete, and points on the tangent space are often thought of as a way to make rigorous the idea of an infinitesimal displacement.

But in many cases we can appeal to prior notions; we learned about tangent lines and tangent planes and stuff in our elementary geometry courses, and these make a perfectly good model for the tangent space to a manifold at a point. In fact, the first time I saw tangent spaces defined in algebraic geometry, it was done in this fashion.

Then, viewing differentials as dual to tangent vectors leads directly to, e.g. in the case of a plane curve, having $(\mathrm{d}x, \mathrm{d}y)$ coordinates for points on the tangent line.