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I've noticed that Spivak, and many other analysis books I read like Munkres, do not use $dx$ when they integrate. Why is that? This is a serious question.

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Spivak writes $dx$, but not every time. For example, he defines the integral of $f$ over $[a,b]$, who he notes as $\int_a^b f$, to be the certain number. He is not "integrating", but rather using a particular notation. If you look into the chapter where he talks about primitives, you'll certainly see he uses $dx$. –  Pedro Tamaroff Mar 3 '13 at 4:21
    
I've Spivak's 2nd edition, 1980, and in many, many cases he writes $\,dx, dt\,$ or whatever... –  DonAntonio Mar 3 '13 at 4:30
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Although your question is fine, I am inclined to think "Why do we write '$dx$' when we integrate?" would be a better question. In other words, surely the burden should lie with explaining this symbol that apparently has no meaning in and of itself. –  Pete L. Clark Mar 3 '13 at 4:36
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OK, looking at my version of Spivak, Chapter 13 is "The Integral", which starts on page 345. On page 364 he explains the relation between his notation and the standard one, and then he uses them interchangeably. What's more, he uses the standard notation with a higher frequency than his notation. –  Pedro Tamaroff Mar 3 '13 at 4:39
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If there is no dummy variable needed, then using $\int_a^b f$ is shorter. As soon as you have a dummy variable, it is necessary to specify what that variable is, so we can write $dx$. This is often better: $\int_a^b x^2\,dx$ vs. $\int_a^b (x\mapsto x^2)$. –  wj32 Mar 3 '13 at 5:02

4 Answers 4

He uses the symbol $$ \int_a^b f $$ because conceptually it makes perfect sense when you first define an integral. It simply denotes the integral of the function $f$ from $a$ to $b$. Later he writes that the symbol $$ \int_a^b f(x) \, dx $$ is also used as the later notation is much easier to deal with when there is a formula for the function (and uses this notation in later chapters like Integration in Elementary Terms). He compares it to limits where the notation $$ \lim_{x \to 2} f(x) $$ is used (since it is a more useful notation) instead of something like $$ \lim_2 f. $$ Also, Spivak mentions that you should think of $dx$ as just a symbol telling you what the variable of integration is and nothing more. This is the same reason my professor was not comfortable for including $dx$ since this symbol has a meaning but it is not really defined until you study differential forms.

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I've always felt that $dx$ has meaning as a reminder of what the Riemann integral's definition is. Just as $\sum\to\int$, $\Delta x_i\to dx$. –  alex.jordan Mar 3 '13 at 20:10
    
@alex.jordan Its fine to think of it like that, but I was talking more about a rigorous definition of $dx$ as a mathematical object. No definition is given in Spivak's Calculus or any other single variable calculus book because you need higher mathematics. So, what you can do is to take $dx$ simply as a notation until defined later. It is rigorously defined as a differential form when studying multivariable calculus, for example in Spivak's Calculus on Manifolds. (By the way, I will study it in next year's calculus course so I still don't know what differential forms are :) ) –  Pratyush Sarkar Mar 3 '13 at 20:44
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To add to the answer from Pratyush, the symbol $f$ represents the function and $f(x)$ represents image of $x$ under $f$. When we do definite integration we deal with integral of a function over some interval and then the variable $x$ and $dx$ are not really necessary. But normally functions are given as expressions and then it makes sense to keep a variable like $x$. –  Paramanand Singh Jul 18 '13 at 17:02

If you're working with an arbitrary function (or one that has already been defined) and the context is understood, some authors will write $\int\!f$ or $\int_a^b\!f$ rather than $\int_a^b\!f(x)\,dx$ to save space. As long as the context is understood, no information is lost. It's technically not wrong to do so; however, when students are being taught to integrate or one is integrating in a more general sense than Riemann integration, it's much safer to write $\int_a^b\!f(x)\,dx$ or $\int_a^b\!f\,d\mu$ to avoid confusion. In more advanced settings, the notations are used interchangeably, but when one is first learning calculus, the notation $\int_a^b\!f(x)\,dx$ helps students in keeping the variable of integration and functions straight ($dx$ reminds you to integrate with respect to $x$). In a first calculus course, many students will not have seen the difference between $f$ (a function) versus $f(x)$ (a number), and seeing the notation $\int\!f$ can be slightly confusing, as the difference between $f$ and $f(x)$ is not entirely clear (even though $\int\!f$ means the same thing as $\int\!f(x)\,dx$).

Historically, $\int_a^b\!f(x)\,dx$ was used as a suggestive notation, where the $\int$ corresponded to the $\sum$ in the Riemann sum, and in fact, it is an elongated s because it is the "sum of infinitely many rectangles with height $f(x)$ and 'infinitely small' width $dx$." Eventually, writers used $x_0,\ldots, x_n$ to denote the points of a partition, and the abbreviation $x_i - x_{i - 1} = \Delta x_i$ became standard. The integral was defined as the limit as the mesh of the partition tended to $0$ (and in other more rigorous ways as well later on), or as $\Delta x_i\to 0$, of the sums $\sum_{i = 1}^n f(x_i)\Delta x_i$. The way each piece of the sum changes to a pice of the integral when the limit is taken is quite suggestive, and pleases many people (not to mention it reminds us of how one actually obtains the Riemann integral). [History taken from footnotes in Spivak]

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I find idea that the $dx$ notation is useful in freshman calculus because students don't know the difference between the function $f$ and the number $f(x)$ to be extremely disturbing...which is not to say that I disagree with it. –  Pete L. Clark Mar 3 '13 at 8:05

Spivak is trying to give the impression of a well-defined, rigorous, step by step development of a theory with solid foundations. Nothing would do more to shake the reader's confidence in this project than appearing to rely on the effective but magical Leibniz $f(x)dx$ notational voodoo. So he uses integrals of functions over subsets $\int f$, not integrals of differential forms $\int f(x) dx$. As a concession to the reality that users of his book may be students in actual calculus classes, he does mention the traditional notation and in the chapter on algebraic rules for integration the exercises are in Leibniz form.

Spivak also avoids Leibniz notation for derivatives, and the language of infinitesimals. He explains the $df/dx$ symbolism and its formal properties, but never relies on it in a proof. Only $f'$ is used. Leibniz notation doesn't have to be non-rigorous, but it creates an air of mystery and this is not consistent with the goals of Spivak's book.

More:

a search through the online part of the 3rd edition

http://books.google.com/books?id=7JKVu_9InRUC&q=leibnizian

finds a statement that "Leibnizian notation" is ambiguous and will be avoided in the main text but illustrated in some of the exercises. This is done in the chapters on derivatives. For integrals it looks like the main text avoids use of $\int f(x) dx$ in general theorems about integration of functions, but allows the $dx$ form when talking about integrals of particular functions, and in the exercises the integrals are mostly in $dx$ form. Spivak seems to regard the Leibniz differential notation as sloppy for derivatives but grudgingly acceptable for integrals.

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I didn't down vote. To make things entirely clear, the use of "give the impression" feels like you're saying that he actually doesn't provide " a well-defined, rigorous, step by step development of a theory with solid foundations." I'll delete the comments (all of them but this one). –  Pedro Tamaroff Mar 3 '13 at 5:58
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@zyx: "So he uses measure-theoretic integrals over subsets, not integrals of differential forms." What are you talking about? In his Calculus book Spivak doesn't discuss either of these things. He develops -- with complete rigor -- the Riemann integral. –  Pete L. Clark Mar 3 '13 at 7:58
    
@PeteL.Clark, Spivak's notation corresponds to the idea of integral over a subset, as is often denoted by an integral sign above (say) X for integration over a set X, but without reference to a variable like $x$ parametrizing the set (as in $f(x) dx$). The upper and lower limits of integration are labels for the set, there is obviously no difference between that and taking $X = [a,b]$. He does not develop measure theory or use the measure theoretic $d\mu$, and I don't think I was claiming that. If you know a better set of words to describe the situation feel free to suggest. –  zyx Mar 3 '13 at 8:01
    
@PeterTamaroff, regardless of whether Spivak's book is completely rigorous (and there are arguments for and against this interpretation), I believe his goal as an author included creating the impression in the reader that a solid theory is being constructed. The absence of this impression when reading a traditional calculus book is part of the rationale for writing a Spivak- or Apostol- style book. Those two were the first widely used books in the United States that present calculus systematically as a body of theory. –  zyx Mar 3 '13 at 8:26
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@PeteL.Clark: if one is writing a calculus book with the agenda to avoid Leibniz notation, but retain the integral sign, there aren't a lot of notational options that don't look like the measure theory integrals, are there? The notation will indicate a function and a set over which to integrate, but no additional information such as a coordinate or parameterization of the subset. And since we are using one fixed measure (the normalized, translation invariant one, whether constructed in the Riemann or Lebesgue way or some other), there will be no $d\mu$. –  zyx Mar 3 '13 at 8:59

Part of the issue here is that there are really two things going on in basic integral calculus:

  • A theory of "definite integrals of functions": given a continuous real valued function $f$ on an interval $[a,b]$ we want to associate a real number, the definite integral $\int_a^b f$.

  • A theory of how to compute these definite integrals using symbolic manipulation (that's what the word "calculus" means). In this setting, we have an expression, which is not the same as a function, and we want to work out a definite integral algorithmically in terms of this expression. This is what Wolfram Alpha does, for example - it manipulates expressions, not functions.

In the second setting, is not always obvious which function is being defined by an expression. For example, it is not at all obvious what $\int_1^3 zc$ would be - what function is $zc$?. In this case, we have to name the variable that is being used to define the function, for example $\int_1^3 zc\,dc$ means that we are integrating the function $f(x) = zx$ from $1$ to $3$, and per convention we are supposed to assume $z$ is a fixed real number. So the calculus formula $\int_1^3 zc\,dc = 4z$ really means: if $z$ is a fixed real number and $f$ is the function on $[1,3]$ with $f(x) = zx$ then $\int_1^3 f$ will equal $4z$.

Historically, nearer the time when calculus was developed, mathematicians such as Johann Bernoulli and Euler used "function" as a synonym for "expression". The routine use of functions that are not defined by particular expressions was not common at that time as it is now.

Modern authors are aware of the distinction, but for various reasons it is not emphasized in textbooks. Texts usually present integral rules in terms of expressions, but prove (or "prove") theorems in terms of functions. To bridge that gap, they sometimes need to indicate which variable is used to define a function from an expression. That is the modern purpose of the "dx" notation.

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