# Dx before the f(x) in integrals

Demystifier
Gold Member
• You cannot take away a loved infinitesimal from physicists.
I would propose that all physicists should learn non-standard analysis, just for the sake of replying to pretentious mathematicians who mock physicists for using infinitesimals.

• Half of them would immediately lose the ability to perform a correct substitution.
Physicists don't do any substitutions anyway. They solve integrals either by looking into a comprehensive math handbook such as Bronstein et al (especially if they are old enough), or put it into Mathematica.

nrqed
Homework Helper
Gold Member
Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?
Well, I have rarely seen multiple integrals in physics written with the ##d^nx## (or whatever variable) written at the far end, it is always written right next to the integral symbol. Therefore it is logical to use the same notation when there is a single integration, no?

Well, I have rarely seen multiple integrals in physics written with the ##d^nx## (or whatever variable) written at the far end, it is always written right next to the integral symbol. Therefore it is logical to use the same notation when there is a single integration, no?
Oh, the question was about integrals in general, not just single integrals.

By analogy with ##\sum_n f_n## I would propose to use a third notation
$$\int_x f(x)$$
The usual mathematical notation is ##\int_Sf## or ##\int_Sfd\mu## if you want to emphasize the measure.

If you have a double integral, then the notation
$$\int_{a}^{b}\int_{c}^{d}f(x,y)dxdy$$
is ambiguous, because it is not clear whether ##x\in[a,b]## or ##x\in[c,d]##. With notation
$$\int_{a}^{b}dx\int_{c}^{d}dy \,f(x,y)$$
there is no risk for such a confusion.
The first notation is not ambiguous unless the text is very poorly written. It is used in many math books and i have never seen anyone, including tones of american undergrad students, be confused by it. Of course the way you've written it ##x\in[c,d]##. The integral sign ##\int## and the differential ##dx## are just like parentheses.The second notation looks very strange to me because surely ##\int_a^bdx=b-a##.

Demystifier
Gold Member
The second notation looks very strange to me because surely ##\int_a^bdx=b-a##.
$$\sum_{n=1}^{10}\sum_{m=1}^{10}f_{nm}\; ?$$
Does it look strange to you because surely ##\sum_{n=1}^{10}=10##?

nrqed and weirdoguy
$$\sum_{n=1}^{10}\sum_{m=1}^{10}f_{nm}\; ?$$
Does it look strange to you because surely ##\sum_{n=1}^{10}=10##?
There is no closing bracket here the ##\int\dots dx## is analogous to ##(\dots)##. To be analogous it needs to be like $$\left(\sum_{m=1}^{10}\right)f_m.$$
A sum is an integral with respect to the counting measure and usually the measure is not explicit in the notation. Strictly the sum $$\sum_{m=1}^{10}f_m$$ is $$\int_{\{1,\dots, 10\}}f(m)dm$$ or if you prefer a different notation for the integral $$\sum_{m=1}^{10}f_mdm.$$ Then of course you could think that $$\sum_{m=1}^{10}dmf_m = 10f_m.$$

By the way, this ##\int(x+1)dx## means integrate the function ##x+1## and you are comfortable to write it as ##\int dx(x+1)##.

What about ##(x+1)^3##? It means cube the function ##x+1##, do you ever write it as ##()^3(x+1)## or as ##{}^3 (x+1)##?

Demystifier
Gold Member
By the way, this ##\int(x+1)dx## means integrate the function ##x+1## and you are comfortable to write it as ##\int dx(x+1)##.

What about ##(x+1)^3##? It means cube the function ##x+1##, do you ever write it as ##()^3(x+1)## or as ##{}^3 (x+1)##?
You view ##dx## as the right bracket, so for you it must be on the right. I view ##dx## as an infinitesimal multiplied with ##f(x)##, so, since multplication is commutative, we have ##dx \,f(x)=f(x)dx##.

For instance, if ##v(t)## is time-dependent velocity, the infinitesimal path is
$$dx=v(t)dt=dt\,v(t)$$
so
$$x=\int dx=\int v(t)dt=\int dt\,v(t)$$

dlgoff, nrqed and weirdoguy
You view ##dx## as the right bracket, so for you it must be on the right. I view ##dx## as an infinitesimal multiplied with ##f(x)##, so, since multplication is commutative, we have ##dx \,f(x)=f(x)dx##.
This was already pointed out, but it doesn't explain the choice. You say they are the same, but in some cases you prefer ##dxf(x)##. I am simply curious why. I guess you wouldn't write that way differential forms? How about integrals not in the sense of Riemann? There the infinitesimals that commute is not very meaningful.

Demystifier
Gold Member
This was already pointed out, but it doesn't explain the choice. You say they are the same, but in some cases you prefer ##dxf(x)##. I am simply curious why.
I think the best explanation why is given in my #18. Your objection in #30 does not make much sense when ##dx## is not viewed as a right bracket.
How about integrals not in the sense of Riemann? There the infinitesimals that commute is not very meaningful.
Could you be more specific about that? What kind of integral do you have in mind? If you talk about Grassmann/Berezin integral, there ##d\eta## cannot be interpreted as an infinitesimal at all.

Demystifier
Gold Member
Perhaps the whole problem stems from the fact that it is sacrilegious for mathematicians to think of ##dx## as an infinitesimal. For if it is not an infinitesimal, then what is it? The only remaining option seems to be that ##dx## is a kind of a right bracket, so it must be on the right. But sloppy physicists do not have problems with thinking of ##dx## as an infinitesimal, so they have a freedom to put it either on the right or on the left. Then they make a final choice by other criteria, such as those in post #18.

Demystifier
Gold Member
There is also a good cognitive reason to prefer ##\int dx\, f(x)## over ##\int f(x)dx##. In plain English, the first would be expressed as "integrate over x the function f", while the second would be "integrate the function f over x". But when one just says "integrate", the first question that comes to one's mind is "integrate over what?", especially if there are many variables involved. So it seems cognitively more natural to say "integrate over x the function f", rather than "integrate the function f over x".

EDIT: Linguistically, "integrate the function f over x" sounds better than "integrate over x the function f". But when I translate it to my native Croatian language, it is no longer the case. How about other languages?

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nrqed
Demystifier
Gold Member
Another thought. Sometimes the order of integration in multiple integrals matters. But if I write this as
$$\int\int f(x,y)dxdy \neq \int\int f(x,y)dydx$$
it looks confusing to me because my first thought is that it implies ##dxdy \neq dydx##, which of course is wrong. On the other hand, if I write this as
$$\int dx\int dy \, f(x,y) \neq \int dy\int dx \, f(x,y)$$
it does not make me confused. Indeed, if I think of ##\int dx## as an operator that may act on a function, then I can interpret the above as non-commutativity of operators
$$\int dx\int dy \neq \int dy\int dx$$
which is quite true.

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fresh_42
Mentor
But when I translate it to my native Croatian language, it is no longer the case. How about other languages?
German doesn't have the strict SPO rule, so you could say:
• I integrate f over x.
• Over x I integrate f.
• f w.r.t.x must be integrated.
• f must be integrated over x.
• I integrate over x the function f.
although some of them sound a bit constructed.

dlgoff
EDIT: Linguistically, "integrate the function f over x" sounds better than "integrate over x the function f".
In polish "integrate the function f over x" sounds better and actually I don't see any other way to say it.

Demystifier
Gold Member
And also when did that start?
So far we said nothing about that. I have just checked out several old physics books and they all use ##\int f dx##. The oldest physics book with ##\int dx\, f## that I found is Schweber (1961). Can someone find an older example?

Could you be more specific about that? What kind of integral do you have in mind?
Well, any integral really that comes from a measure. For example how do you think of Lebesgue's integral as an infinite sum of infinitesimals?
EDIT: Linguistically, "integrate the function f over x" sounds better than "integrate over x the function f". But when I translate it to my native Croatian language, it is no longer the case. How about other languages?
For me nether sounds right. You don't integrate over x, you integrate with respect to x and over a set. By the way how is it in Croatian?
it looks confusing to me because my first thought is that it implies ##dxdxdy \neq dydx##, which of course is wrong.
One would have to say what meaning one puts in this notation otherwise you cannot simply say that it is wrong. For example if these are suppose to be differential forms, then certainly ##dx\wedge dy \neq dy\wedge dx##.
So far we said nothing about that. I have just checked out several old physics books and they all use ∫fdx∫fdx\int f dx. The oldest physics book with ∫dxf∫dxf\int dx\, f that I found is Schweber (1961). Can someone find an older example?
Finally something towards my actual questions. How did Dirac write his integrals? Somewhere I saw a statement that Leibniz was using both notations! In any case there must have been a switch, I still want to know why. Also every one who writes it the "physicists" way must at some point in his life switch because it is more likely that he started in school or university studying integrals in a math course.

Demystifier
Gold Member
Finally something towards my actual questions. How did Dirac write his integrals?
He did it like a mathematician.

Also every one who writes it the "physicists" way must at some point in his life switch because it is more likely that he started in school or university studying integrals in a math course.
When I reread my undergraduate textbooks, it seems that I was first exposed to the physicist-like notation at the 3rd year, in Jackson's Classical Electrodynamics.

Demystifier
Gold Member
By the way how is it in Croatian?
Integral po x od funkcije f.

martinbn
fresh_42
Mentor
So far we said nothing about that. I have just checked out several old physics books and they all use ##\int f dx##. The oldest physics book with ##\int dx\, f## that I found is Schweber (1961). Can someone find an older example?
R.Courant / D.Hilbert, 1924, kept it consequently at the end. The first appearance I have found was in:
E. Madelung, 1950, Die Mathematischen Hilfsmittel des Physikers (The Mathematical Tools of the Physicist)
but only for those integrals with long integrands like ##F(t,v) = \int dv \int dt \varphi(x,t)e^{\{\,\ldots\,\}}##

DrDu
By analogy with ##\sum_n f_n## I would propose to use a third notation
$$\int_x f(x)$$
This form is used in the theory of differential forms!

Demystifier
Gold Member
This form is used in the theory of differential forms!
I think in differential forms it is more like
$$\int_V f$$
where ##V## is the domain of integration, not the variable of integration.

dlgoff and fresh_42
Demystifier