# Dx before the f(x) in integrals

Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?

Demystifier

Psinter

I have never seen that before.

https://math.stackexchange.com/ques...in-usage-for-dx-before-or-after-the-integrand

Although it doesn't say when it started.

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Mentor
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Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?
It's a matter of taste. E.g. if we have especially long integrands with multiple variables and constants, it can be very helpful to denote the integration variable first. My guess is, that some physicists started with it for exactly this reason: write down the "unnecessary" first and concentrate on the essential part.

Demystifier and symbolipoint
Homework Helper
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Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?

In QM Dirac notation, it is a more natural extension of the representation of vector/state for a countable basis, ##|n \rangle##:

$$|\alpha \rangle = \sum_n |n \rangle \langle n | \alpha \rangle$$
For a continuous basis, ##|x \rangle##, this becomes:
$$|\alpha \rangle = \int dx |x \rangle \langle x| \alpha \rangle$$
Where we have the identity:
$$\sum_n |n \rangle \langle n | = I$$
and
$$\int dx |x \rangle \langle x| = I$$

Stephen Tashi and Demystifier
Staff Emeritus
Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##?

Says who? I'm a physicist, and I don't write it like that most of the time. I've seen mathematicians wrote dx first many times.

And why does this even merit a question? Isn't this no different than asking why someone prefers the color yellow instead of blue?

Zz.

In QM Dirac notation, it is a more natural extension of the representation of vector/state for a countable basis, ##|n \rangle##:

$$|\alpha \rangle = \sum_n |n \rangle \langle n | \alpha \rangle$$
Yes, this is the same. Mathematicians wouldn't put basis vectors first and the numbers second. It would be ##\sum a_i\vec{e}_i##, not ##\sum\vec{e}_i a_i##.

Demystifier
Says who? I'm a physicist, and I don't write it like that most of the time. I've seen mathematicians wrote dx first many times.
Ok, I thought it was obvious that I meant some physicists not all. But you are right I was sloppy.

And why does this even merit a question? Isn't this no different than asking why someone prefers the color yellow instead of blue?
Well, at some point someone must have started doing it. I might be wrong but I think the other notation have been around for a while before some, not all, started using the differential first notation. So there must have been a reason.

Also I have never seen seen it in introductory calculus/analysis textbooks. My guess is that physicists are exposed to it later on. It is puzzling to me that it is used.

Staff Emeritus
Well, at some point someone must have started doing it. I might be wrong but I think the other notation have been around for a while before some, not all, started using the differential first notation. So there must have been a reason.

Also I have never seen seen it in introductory calculus/analysis textbooks. My guess is that physicists are exposed to it later on. It is puzzling to me that it is used.

Again, why is this "puzzling"? Unless you think that these things do not commute, does it matter that I write the product of A and B as BA instead of AB?

It is also a matter of typesetting style. Maybe some publishers or journals have a standard where the integration variables are written first. This is often useful if the integrand itself is a long, complicated function.

Once again, isn't this a matter of personal preference? Should I need to conform to liking the same color as you do?

Zz.

fresh_42
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Yes, this is the same. Mathematicians wouldn't put basis vectors first and the numbers second. It would be ##\sum a_i\vec{e}_i##, not ##\sum\vec{e}_i a_i##.

Yes, exactly, and the Dirac notation doesn't quite work the mathematician's way round! It's the same with the inner product being linear in the second argument, rather than the first.

Yes, exactly, and the Dirac notation doesn't quite work the mathematician's way round! It's the same with the inner product being linear in the second argument, rather than the first.
Yes, and it is convenient that way. For the integral it is not, at least not any more. Are you saying that it is because of Dirac's notations? If you've been writing your integrals one way for a while there has to be a good reason to change later on, no?

Homework Helper
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Yes, and it is convenient that way. For the integral it is not, at least not any more. Are you saying that it is because of Dirac's notations? If you've been writing your integrals one way for a while there has to be a good reason to change later on, no?

All I know is that I started doing the integrals the other way round when I was learning QM Dirac notation. I've no idea whether Dirac started it. That's the only time I do it that way. Any other time I put the ##dx## at the end.

It's the same with vectors, it's only in QM that I write things back to front.

I have given it a little more thought. Here is a speculation. For some people, at least for me, the notation ##\int \dots dx## serves as parentheses to enclose the expression to be integrated, so it is natural to put it that way. Also if one does more abstract integration, or integration on groups it would seem awkward otherwise. Of course just ##\int\dots## would be fine. For other people it may be more important, at least in some cases, or more natural to think of the integral as an operator. It takes a function and it produces something else, a number, a function so on. Then it is more natural to write it as an operator ##\int dx\dots## acting on functions.

Demystifier
Jehannum
Says who? I'm a physicist, and I don't write it like that most of the time. I've seen mathematicians wrote dx first many times.

And why does this even merit a question? Isn't this no different than asking why someone prefers the color yellow instead of blue?

Zz.

Of course it "merits" a question. It elicited several sensible answers.

symbolipoint
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By analogy with ##\sum_n f_n## I would propose to use a third notation
$$\int_x f(x)$$

PeroK
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By analogy with ##\sum_n f_n## I would propose to use a third notation
$$\int_x f(x)$$
Makes sense, and before there will be complaints about the location for the boundaries:
$$\int_{x \in (a,\infty]} f(x)$$

Gold Member
Makes sense, and before there will be complaints about the location for the boundaries:
$$\int_{x \in (a,\infty]} f(x)$$
To make the analogy with sums complete, how about
$$\int_{x =a}^{b} f(x) \;\;?$$

Gold Member
Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?
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.

jedishrfu and nrqed
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To make the analogy with sums complete, how about
$$\int_{x =a}^{b} f(x) \;\;?$$
I guess, because I like to write the sums as well as sum over instead of from to.

And if the integration order is arbitrary, we can even write
$$\int_{x_1}^{x_2}\, \int_{y_1}^{y_2} \, \int_{z_1}^{z_2} \,f(x,y,z) = \iiint_\stackrel{\stackrel{x \in [x_1,x_2]}{y \in [y_1,y_2]}}{\stackrel{z \in [z_1,z_2]}{}}\,f(x,y,z) = \int_{(x,y,z)\in [x_1,x_2]\times [y_1,y_2] \times [z_1,z_2]}\,f(x,y,z)$$
in which case the term volume gets a complete new feeling!

Demystifier
Gold Member
By analogy with ##\sum_n f_n## I would propose to use a third notation
$$\int_x f(x)$$
Let me also note that Schiff in his quantum mechanics textbook uses the same notation
$${\large\sf S}_k f_k$$
for both sums and integrals.

Gold Member

And if the integration order is arbitrary, we can even write
$$\int_{x_1}^{x_2}\, \int_{y_1}^{y_2} \, \int_{z_1}^{z_2} \,f(x,y,z) = \iiint_\stackrel{\stackrel{x \in [x_1,x_2]}{y \in [y_1,y_2]}}{\stackrel{z \in [z_1,z_2]}{}}\,f(x,y,z) = \int_{(x,y,z)\in [x_1,x_2]\times [y_1,y_2] \times [z_1,z_2]}\,f(x,y,z)$$
in which case the term volume gets a complete new feeling!
Isn't that a proof that physicist's notation is better?

Mentor
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Isn't that a proof that physicist's notation is better?
I might have agreed, if it wouldn't have happened, that I read this thread here in parallel

Gold Member
I might have agreed, if it wouldn't have happened, that I read this thread here in parallel

Mentor
2021 Award
• You cannot take away a loved infinitesimal from physicists.
• Half of them would immediately lose the ability to perform a correct substitution.

Demystifier
Gold Member
The greatest notation master among physicists is DeWitt, who writes e.g.
$$\int d^4x \sum_{\mu=0}^3 j_{\mu}(x)A^{\mu}(x)$$
simply as
$$j_kA^k$$

fresh_42
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.

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##.

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)##?

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