Why Do Physicists Write Integrals as ##\int dx f(x)##?

[martinbn]

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

 

[Psinter]

I have never seen that before.

Found a quick answer (with another linked question and answer):

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

Although it doesn't say when it started.

 

[fresh_42]

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.

 

[PeroK]

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
$$\int dx |x \rangle \langle x| = I $$

 

[ZapperZ]

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.

 

[martinbn]

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 $\sum\vec{e}_i a_i$.

 

[martinbn]

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.

 

[martinbn]

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.

 

[ZapperZ]

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.

 

[PeroK]

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.

 

[martinbn]

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?

 

[PeroK]

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.

 

[martinbn]

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.

 

[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.

 

[Demystifier]

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

 

[fresh_42]

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:
$$\int_{x \in (a,\infty]} f(x)$$

 

[Demystifier]

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
$$\int_{x =a}^{b} f(x) \;\;?$$

 

[Demystifier]

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.

 

[fresh_42]

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]

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.

 

[Demystifier]

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?

 

[fresh_42]

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

 

[Demystifier]

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

Well, in this thread physicists are silly, but this thread is not about notation.

 

[fresh_42]

There are two major disadvantages with your proposal:

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

 

[Demystifier]

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
$$j_kA^k$$

 

[Demystifier]

• 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]

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?

 

[martinbn]

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.

 

[martinbn]

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.

 

[martinbn]

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]

The second notation looks very strange to me because surely $\int_a^bdx=b-a$.

How about
$$\sum_{n=1}^{10}\sum_{m=1}^{10}f_{nm}\; ?$$
Does it look strange to you because surely $\sum_{n=1}^{10}=10$?

 

[martinbn]

How about
$$\sum_{n=1}^{10}\sum_{m=1}^{10}f_{nm}\; ?$$
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 $$\sum_{m=1}^{10}dmf_m = 10f_m.$$

 

[martinbn]

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]

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
$$x=\int dx=\int v(t)dt=\int dt\,v(t)$$

 

[martinbn]

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]

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]

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]

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?

 

[Demystifier]

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.

 

[fresh_42]

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.

 

[weirdoguy]

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]

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?

 

[martinbn]

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]

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]

By the way how is it in Croatian?

Integral po x od funkcije f.

 

[fresh_42]

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]

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.

 

[Demystifier]

For example how do you think of Lebesgue's integral as an infinite sum of infinitesimals?

If Riemann's integral is a sum of infinitesimal vertical strips, then Lebesgue's integral is a sum of infinitesimal horizontal strips. See e.g. the picture in https://en.wikipedia.org/wiki/Lebesgue_integration .

 

[fresh_42]

If Riemann's integral is a sum of infinitesimal vertical strips, then Lebesgue's integral is a sum of infinitesimal horizontal strips. See e.g. the picture in https://en.wikipedia.org/wiki/Lebesgue_integration .

... or https://www.physicsforums.com/insights/omissions-mathematics-education-gauge-integration/