6
$\begingroup$

A few of us over on MITx have noticed that $\int f(x) dx $ is appearing as $\int dx f(x)$.

It's not the maths of it that worries me. It's just I recently read a justification (analytical?) of the second form somewhere but can't recall it or where I saw it.

Can anyone give me a reference?

  • 2
    It's just notation.2012-06-21
  • 1
    It's just a different notational convention (common in physics).2012-06-21
  • 0
    Yes, but there was a reason for it being a **preferred** notation...2012-06-21
  • 1
    One "justification" I've heard is that if you consider integration as a function taking, say, integrable functions to real number, then writing $\int \mathrm{d}x f$ sort of looks like "applying the function $\int \mathrm{d}x$ to $f$". I don't like the argument myself, but it's one I've heard at least.2012-06-21
  • 0
    I've "reasoned" similarly to myself: $\int dx$ is an operator; putting the dx after the operand function is just using it as a bracket... But I don't think that's the justification I saw.2012-06-21
  • 1
    You can also think of integrating as a kind of sum over these infinitesimal products $f(x)dx$. So then interchanging the factors seems natural, i.e. then both orders are regarded as equivalent. Especially if you have many different variables and dimensions it is prudent to see at the beginning with respect to which you are integration. So you don't first have to match all integral signs with their respective $d$s.2012-06-21
  • 0
    @canaaerus, that's a good, practical point I hadn't thought of.2012-06-21
  • 1
    The argument I heard and which is quite convincing is that in physics $f(x)$ can have very long form. So in order not to forget about $dx$ we write it first :)2012-06-21
  • 0
    Brilliant! You're not a physicist are you? :-)2012-06-21
  • 0
    @AppliedImagination actually I study physics :)2012-06-21
  • 2
    See http://math.stackexchange.com/questions/128108.2012-06-21
  • 1
    see also: http://math.stackexchange.com/q/52302012-06-21
  • 1
    Thanks, t.b. and joriki. I'd turned up these posts, but neither of them or their answers give a rationale for preferring one version or another, and that's what I'm after.2012-06-21

1 Answers 1