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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?

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    $\begingroup$ I'd tend to agree with Spivak in this. I'd just remark to students that the symbol serves us to distinguish with respect to which variable we're integrating. So, in $\;\int \left(3x^{y^3\cos z}\right) dx\;$ , both $\;y,\,z\;$ are taken as constants, and $\;x\;$ is the variable. That's all...and that's a lot, too. $\endgroup$ – DonAntonio Aug 24 '18 at 21:31
  • $\begingroup$ There is a thing called a differential and one of the ways to notate the differential of $x$ is $dx$, but that doesn't mean that every $dx$ you ever see will be a differential. $\endgroup$ – David K Aug 24 '18 at 21:41
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    $\begingroup$ Until one gets to a rigorous multivariable analysis setting and learns differential forms, it's exceedingly difficult to make any rigorous definition of a "differential," and even less so of "infinitesimal." Spivak is right that it's a formal symbolic thing at the level of single-variable calculus, even if physicists and engineers tend to think of "infinitesimal" changes in the variables. But he's not writing an engineering-style textbook here ... :) $\endgroup$ – Ted Shifrin Aug 24 '18 at 21:43
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    $\begingroup$ Even in single-variable calculus the differential plays an important role. It tells us over which variable the integration takes place. This also allows one to adjust the variable. For example if we integrate $f(x) * x dx$, we may prefer to write this as $f(x) d(x^2)/2$. This is a standard operation, that would be difficult to explain or understand if the differentials were dropped from the expression. $\endgroup$ – M. Wind Aug 25 '18 at 16:38
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    $\begingroup$ Furthermore in science integration usually takes place over a variable such as time or frequency or a spatial dimension. In such cases the differential $dx$ has a physical dimension. If one would omit the differential from the expression there could be confusion over the dimension of certain terms. $\endgroup$ – M. Wind Aug 25 '18 at 17:02
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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 $f : \mathbb{R} \to \mathbb{R}$

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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).

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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.

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