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martinbn

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- Thread starter martinbn
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martinbn

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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.

- #3

fresh_42

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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.

- #4

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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

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

- #5

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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.

- #6

martinbn

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

$$|\alpha \rangle = \sum_n |n \rangle \langle n | \alpha \rangle $$

- #7

martinbn

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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.

- #8

martinbn

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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?

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.

- #9

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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.

- #10

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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.

- #11

martinbn

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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?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.

- #12

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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.

- #13

martinbn

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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.

- #15

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By analogy with ##\sum_n f_n## I would propose to use a third notation

$$\int_x f(x)$$

$$\int_x f(x)$$

- #16

fresh_42

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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

$$\int_x f(x)$$

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

- #17

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To make the analogy with sums complete, how aboutMakes sense, and before there will be complaints about the location for the boundaries:

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

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

- #18

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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.

- #19

fresh_42

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I guess, because I like to write the sums as well asTo make the analogy with sums complete, how about

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

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!

- #20

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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

$$\int_x f(x)$$

$${\large\sf S}_k f_k$$

for both sums and integrals.

- #21

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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!

- #22

fresh_42

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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?

- #23

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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

- #24

fresh_42

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

- #25

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$$\int d^4x \sum_{\mu=0}^3 j_{\mu}(x)A^{\mu}(x)$$

simply as

$$j_kA^k$$

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