In old mathematics books, I see a lot of notations like $\int_{0}^{x} f(x) dx$. For example, Courant-Hilbert: Methods of mathematical physics. However, when I wrote it in this site, it was sometimes edited like $\int_{0}^{x} f(t) dt$.
Should the notation $\int_{0}^{x} f(x) dx$ be frowned upon?
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16It shouldn’t just be frowned on: $\int_0^x f(x)dx$ is simply wrong. – 2012-09-02
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7Although the notation $\int_0^x f(x)\,dx$ is fine from a logical point of view, I have always avoided it in teaching. Just as I would never write $\forall x(F(x)\land\forall x G(x))$. – 2012-09-02
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11In most cases it's kind of a nit-picky thing. "Everyone knows" what is meant by $\int_0^x f(x)\,dx$, but there is something unsatisfactory about $x$ pulling double duty. If a student uses the same variable as a dummy variable and as a limit, I tend to overlook it. But in a textbook, it seems kind of shoddy. Why not use a different symbol for total clarity? It's not like you're going to pay extra for using a different letter. – 2012-09-02
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13Sometimes I wonder why we don't just write it as $\int_0^x f$. – 2012-09-02
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1@RahulNarain : If I write "This leads us to consider the integral $\int_{-\infty}^x \exp(-u^2/2)\,du$.", I can think of some reasons why I don't just write what you suggest. – 2012-09-02
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2@Rahul: We sometimes do. But dummy variable notation is too convenient to discard; e.g. it lets us write things like $\int_0^x t^2 \, dt$ without great pain. Sure, you could define $f(t) := t^2$ first so you can write $\int_0^x f$, but you haven't gotten rid of the dummy variable, just shifted it around. – 2012-09-02
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0@BrianM.Scott Could you explain why it's simply wrong? – 2012-09-02
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0@Makoto: Mathematical grammar is often designed to forbid such ambiguity: you're not allowed to introduce $x$ as a dummy variable into any context where $x$ already appears. – 2012-09-02
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7Because it violates the very strong notational convention against using one variable name to refer to two different variables in the same expression. It also subverts one of the main functions of mathematical notation, which is to **facilitate** understanding. – 2012-09-02
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0@BrianM.Scott As I said, it was used often in old mathematics books. So it's rather traditional notational convention. – 2012-09-02
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4It might be a traditional notational convention, but that does not prevent it from being simply wrong by modern standards. – 2012-09-02
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3@Michael: Indeed, what we need is more [pointfree style](http://www.haskell.org/haskellwiki/Pointfree) in mathematics. $\int_{-\infty}^x \exp\circ(-\frac12)\circ\operatorname{square}$, right? :) – 2012-09-03
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0@RahulNarain : I suspect I can think of some reasons to avoid your point-free notation, but it's going to take more work to express it cogently. One reason would be that if $f(u)$ is in meters per second and $du$ in seconds, then in the expression $f(u)\,du$, the seconds cancel and we get the right units. But that's not the only reason. – 2012-09-03
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0@Michael, I know, I know, I'm just fooling about at this point. – 2012-09-03
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0@Makato Because basically in that case your variable $x$ runs from $0$ to $x$. – 2012-09-03
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0@AndréNicolas where is $\forall x(F(x)\land\forall x G(x))$ used?, or in which context? I really have no idea what you are refering to, because of my lack of knowledge. – 2012-09-03
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0@MaoYiyi: If you are not familiar with notation from symbolic logic, that part of the comment would not mean much to you. It is an example of using $x$ for two different purposes in the same sentence. It turns out that by the rules of logic it is legal. But as a practical matter it is not a good idea. – 2012-09-03
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0@BrianM.Scott I don't think just saying it's simply wrong is very persuasive. – 2012-09-03
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2I gave you a reason: it violates the currently accepted notational conventions. – 2012-09-03
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0You may also see $a-x.b-x$ in some old books for $(a-x)(b-x)$; I think that I remember seeing it in Cayley’s *An Elementary Treatise on Elliptic Integrals*, for instance. That doesn’t change the fact that it’s simply wrong today. Notational standards, like standards of rigor, can change over time. – 2012-09-03
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0@BrianM.Scott I'm talking about the 20th century mathematics. It's not that old. – 2012-09-03
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2I started reading calculus texts in the late 1950s. I have never seen expressions like $\int_0^xf(x)\,dx$ in such texts. I doubt that this confusing notation has been in common use in the last $60$ years, at least in the U.S. And it really doesn’t matter: by currently accepted standards it’s at best very confusing and at worst simply wrong. If you understand this fact, I don’t see what your point is. If you don’t, you need to learn it. – 2012-09-03
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0@BrianM.Scott As I wrote, Courant-Hilbert's book(1953) adopted such notations. Tenenbaum-Pollard's Ordinary differential equations(1963) also adopted such notations. – 2012-09-03
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0@BrianM.Scott H.M.Edwards Advanced calculus(1993) p.250 – 2012-09-03
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0@AndréNicolas thanks for letting me know the branch of math. – 2012-09-04
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1@OldJohn Could you explain why it's simply wrong by modern standards? It seems to me that just saying so without an explanation is not very persuasive. – 2012-09-24