# Notational change with Integrals

A little over 50 years ago I took my first Calculus class and learned the conventional form of an integral as: $$\int f(x)\,\, \textrm{d}x$$ That is, the integral sign (definite or indefinite) followed by the function being integrated followed by the integration variable as $\textrm{d}x$ (in this example).

However, during retirement I decided a self-teaching adventure into Quantum Field Theory and also General Relativity and other topics of mathematical physics. Reading many different papers and text books on these subjects I find a lot of authors using this variation of the integral: $$\int \textrm{d}x\,\,f(x)$$ Where the integral sign is immediately followed by designating the variables to be integrated. Other examples include: $$S = \int_{t_1}^{t_2}\textrm{d}^4q\,\,\mathcal{L}(\phi,\dot{\phi}\,;t)$$ For the action integral of a field described by Lagrangian Density function or an example from General Relativity, $$S=\int \textrm{d}^4 x\sqrt{-g}\,\,g^{\mu \nu}\,\,R_{\mu \nu}\,(\,\Gamma\,)$$ Now that I am used to seeing and using this notation, I much prefer it over the notation I learned in Calculus classes and have used ever since in work and play. I like the idea of seeing the integral, its limits, and the variables of integration all described at once before considering the function to be integrated.

My question, is this notation recognized by others, in particular mathematicians as I myself have only seen it used in various papers and texts of topics in mathematical physics. Also, who used it first and were they motivated by the same idea that I personally find as an improvement -- that is, it just reads better.

• It is just a question of convenience, has no deep meaning. Both notations are used. May 4 '17 at 18:18
• You find the notation $\int\;dx\;f(x)$ only in physics. Not in mathematics anymore. Those guys use it out of tradition. Some say it is better in some way, but others disagree. May 5 '17 at 12:50
• Also, especially in a single variable, or if variables are not "separated", it is both legitimate and efficient to just write $\int f$, or $\int_a^b f$, since the "variable of integration" is just a dummy (and any measure other than the usual Lebesgue measure should have been already specified...) Even in more than one variable, using an arrow disambiguates better: $t\to \int f(t,-)$ is efficient. Jan 17 '18 at 23:25

Calculus was originally formulated in terms of infinitesimals. Hundreds of years later, a second formulation was found in terms of limits. There were originally some doubts about whether the version using infinitesimals was logically OK, but these doubts were cleared up by Robinson and others ca. 1961.

The Leibniz notation $\int f(x) dx$ was invented in the earlier period, so in this notation, $dx$ a notation for an infinitesimal. You can imagine a Riemann sum with the narrow rectangles having an infinitesimal width $dx$. The thing inside the integral sign is the width of one such rectangle: its height $f(x)$ multiplied by its width $dx$. Since multiplication is commutative, we have literally $f(x)dx=dxf(x)$.

Some math teachers today still tell their students that the $dx$ is just punctuation, or only functionsas a statement of what variable is being integrated with respect to. This could be because they are afraid their students will be confused by talk about infinitesimals, or because the teachers themselves don't know that any concerns about the logical issues with infinitesimals have been cleared up.

My question, is this notation recognized by others, in particular mathematicians as I myself have only seen it used in various papers and texts of topics in mathematical physics. Also, who used it first[...]

The notation was first used by Leibniz, with the factors written in whichever order was convenient. It is universally recognized, with the factors in either order, by people who understand the above historical and mathematical facts.

• Yes, I understand and agree with your statements above and I knew them when I posed my question. I also know of Robinson's Hyperreal numbers and his infinitesimal approach to Calculus (I own a copy of Keisler's text "Foundations of Infinitesimal Calculus"). But, that is not the answer I was hoping to find. I have a rather elaborate library covering both Mathematics and Physics and not a single Math book uses the notation of the $dx\,f(x)$ order. Sure, it is OK to use them in any order but tradition seems to show Mathematicians have always chosen the one order. May 5 '17 at 15:06
• So, physicists decided to change the order and that is the essence of my question, when and why was that decision made. May 5 '17 at 15:07
• @K7PEH: So, physicists decided to change the order I think you've leaped to the wrong conclusion. I think the order was free originally, then mathematicians stopped liking infinitesimals, and they started thinking of the dx as punctuation and always writing it last. Meanwhile physicists and engineers just kept on doing what they'd been doing for hundreds of years, which was writing the factors in either order, freely.
– user466
May 5 '17 at 21:42
• I think Ben is right. Mathematicians started thinking of $\int$ and $dx$ as open and close parentheses, and always writing $\int f(x)\,dx$, whereas physicists started thinking of $\int dx$ as an operator, and always writing $\int dx \;f(x)$. May 5 '17 at 21:47
• @GeraldEdgar --- your concise comment above is the closest to the kind of answer that seems quite plausible to me since I started to think of the operator idea as well which is why the physics notation appeals to me. May 8 '17 at 18:27

$f(x)dx$ is a product such that commutativity is preserved (like the differential quotient as the limit of a sequence of fractions maintains the notation as a fraction). The "physical" notation spares you some parentheses when integrating over different variables with different ranges: $\int_a^b dx\int_c^d dyf(x, y)$. In $\int_a^b \int_c^d f(x, y)dxdy$ the ranges are less clear. This notation is not only used in physics. See for instance p. 270 of this mathematics text book:https://www.amazon.de/Mathematik-ersten-Semester-Gruyter-Studium/dp/3110377330/ref=pd_cp_14_1?ie=UTF8&refRID=1TB0TDD9MB9DAA735D6K