martinbn
Science Advisor
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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.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##:Why do physicists like to write ##\int dx f(x)## instead of ##\int f(x) dx##? And also when did that start?
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.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?
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?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, 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, 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.
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.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?
Of course it "merits" a question. It elicited several sensible answers.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