- #1
martinbn
Science Advisor
- 2,854
- 1,181
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.Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?
Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?
Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##?
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##.In QM Dirac notation, it is a more natural extension of the representation of vector/state for a countable basis, ##|n \rangle##:
$$|\alpha \rangle = \sum_n |n \rangle \langle n | \alpha \rangle $$
Ok, I thought it was obvious that I meant some physicists not all. But you are right I was sloppy.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.
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.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.
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, 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?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?
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.
Makes sense, and before there will be complaints about the location for the boundaries:By analogy with ##\sum_n f_n## I would propose to use a third notation
$$\int_x f(x)$$
To make the analogy with sums complete, how aboutMakes sense, and before there will be complaints about the location for the boundaries:
$$\int_{x \in (a,\infty]} f(x)$$
If you have a double integral, then the notationWhy do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?
I guess, because I like to write the sums as well as sum over instead of from to.To make the analogy with sums complete, how about
$$\int_{x =a}^{b} f(x) \;\;?$$
Let me also note that Schiff in his quantum mechanics textbook uses the same notationBy analogy with ##\sum_n f_n## I would propose to use a third notation
$$\int_x f(x)$$
Isn't that a proof that physicist's notation is better?
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!
I might have agreed, if it wouldn't have happened, that I read this thread here in parallelIsn't that a proof that physicist's notation is better?
Well, in this thread physicists are silly, but this thread is not about notation.I might have agreed, if it wouldn't have happened, that I read this thread here in parallel
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.
- You cannot take away a loved infinitesimal from physicists.
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.
- Half of them would immediately lose the ability to perform a correct substitution.
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?Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?
Oh, the question was about integrals in general, not just single integrals.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?
The usual mathematical notation is ##\int_Sf## or ##\int_Sfd\mu## if you want to emphasize the measure.By analogy with ##\sum_n f_n## I would propose to use a third notation
$$\int_x f(x)$$
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##.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.
How aboutThe second notation looks very strange to me because surely ##\int_a^bdx=b-a##.
There is no closing bracket here the ##\int\dots dx## is analogous to ##(\dots)##. To be analogous it needs to be like $$\left(\sum_{m=1}^{10}\right)f_m.$$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##?
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##.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)##?
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.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##.