Why does Spivak say "$dx$ (in $\int f(x) dx$) has no meaning in isolation" in his Calculus textbook?

Question

I was under the impression that the $dx$ in $\int f(x) dx$ is called the differential and represents an infinitesimal change in $x$. However, at the bottom of p. 264 in Spivak's Calculus (4th ed.), the author writes

"The symbol $dx$ has no meaning in isolation, any more than the symbol $x \rightarrow$ has any meaning; except in the context $\lim\limits_{x \rightarrow a} f(x)$."

He also states on the next page that for $\int x^2 dx$,

"The entire symbol $x^2 dx$ may be regarded as an abbreviation for: the function $f$ such that $f(x) = x^2$ for all $x$."

Upon looking at the appendix, there are no mentions of the word "differential" in the book. However, he makes use of them later in the book while describing the integral substitution formula using the equations \begin{equation} \begin{split} u &= g(x),\\ du &= g'(x)dx \end{split} \end{equation} and \begin{equation} \begin{split} x &= g^{-1}(u)\\ dx &= (g^{-1})'(u)du. \end{split} \end{equation}

Is anyone familiar with the reasoning behind this seemingly deliberate omission?

Answer 1

So, formally you can't really speak of "infinitely small" quantities (unless you're gonna talk about the hyperreal numbers, but that's a whole different story). This idea does carry the intuition, and before calculus was set on firm footing this was how people thought about it.

When Spivak talks about integration by substitution, the steps you mention serve merely as notational shorthand, and as everything is being done formally, there is no reference made to infinitesimal changes in quantity. It's fine right now to just write $\int_a^b f$, though it is convenient and later on it will actually become important (specifying which variable/measure you integrate against, and later differential forms will give actual meaning to the $dx$ business).

Answer 2

Spivak‘s claim is probably related to the particular use of a differential like $dx$ as part of the composite that is an integral, the other parts being the integral $\int$, integrand expression and, optional, the specification of the domain of integration.

The practical Leibniz Kalkül, which is also used in the context of integration, you cite it towards the end, will involve more than one differential. E.g. $d(fg)=(df)g+f(dg)$.

But in other context that claim seems not valid to me. E.g. for an isoline of a multi variable function $F$ it is custom to characterize it having $dF=0$ along it.

Answer 3

IIRC, if you investigate the book thoroughly, you'll notice he has only ever defined the integral for (suitable) transformations $f : \mathbb{R} \to \mathbb{R}$. Thus, if there was a function such as $g(x, y)$, it would be typed as: $g : \mathbb{R} \to \mathbb{R} \to \mathbb{R}$ and the current definition he supplies does not make any sense for. Hence, you could not simply write $\int g$.

Furthermore, as he mentions in the book, $\int$ is a transformation that with the following type signature:

$$\int : \mathbb{R} \to \mathbb{R} \to \mathbb{F} \to \mathbb{R}$$

where the first two $\mathbb{R}$ are the limits of integration, and the $\mathbb{F}$ is the integrable transformation. Finally, $\mathbb{R}$ is the value. Hence, if you eliminate a lot of the sugaring introduced, what we normally write as:

$$\int_a^b f(x) dx$$

Really becomes:

$$\int(a)(b)(f)$$

As, in the case of Spivak, $f : \mathbb{R} \to \mathbb{R}$, it is (surprisingly!) superfluous to even mention the variable of integration since it is required that there is always precisely one variable of integration¹. Furthermore, there is no mention of $dx$. Thus, if you are trying to convince your computer of a theorem and use $dx = blah$ arguments, the computer will throw an error at you saying that $dx$ is not defined.

In fact, $\int_a^b$ is sugar for $\int(a)(b)$, however, I can't think of a case you could abuse this notation.

^{1 Even a constant function such as $f(x) = c$ still has the signature $f : \mathbb{R} \to \mathbb{R}$}