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Is there an agreed way of expressing undirected integration in formulas?
my idea of doing so would be to use the absolute value of the differential

$$\int_a^b f(x)|dx| = \int_b^a f(x)|dx|$$

but I would like to get some feedback before actually using it.

A typical situation, in which undirected integrals are required, is integrating force along a path.

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    $\begingroup$ How about $\int_{[a,b]}f(x)dx$ with the convention that $[a,b]=[b,a]$? I find it more natural to have undirected intervals than undirected integrals. $\endgroup$ Aug 28, 2014 at 8:34
  • $\begingroup$ @JoonasIlmavirta I see the problem, that [a,b] commonly denotes a closed interval and, integrating over a closed interval can also be direction dependent $\endgroup$ Aug 28, 2014 at 9:29
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    $\begingroup$ as a physicist, I would denote integration of a force along a path by a line integral, $\int_{a}^{b} \vec{F}\cdot d\vec{l}$ $\endgroup$ Aug 28, 2014 at 9:32
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    $\begingroup$ @ManfredWeis, I was thinking of the interval as a set with no additional structure. The Lebesgue integral is naturally taken over a set (without orientation). But if the thing you want to integrate is a vector field or a differential form (or can be conveniently interpreted as such), I would suggest using the related notation. Carlo Beenakker's comment is a good example. $\endgroup$ Aug 28, 2014 at 11:54
  • $\begingroup$ I guess we should think of a situation where $x$ is a function of another variable, and $f$ could go back and forth through parts of $[a,b]$ several times? $\endgroup$ Aug 28, 2014 at 21:11

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I quite like $f(x)|dx|$. Namely, $f(x)dx$ is a 1-form, whereas $f(x)|dx|$ is a density: In differential geometry densities are used to integrate on non-orientable manifolds. They are sections of the line bundle with transition functions $|\det(d (u\circ v^{-1}))|$, from the chart $u$ to the chart $v$. Also useful are half-densities, which form a natural pre-Hilbert space without needing a measure.

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On the other hand, in my opinion, the advantage of a standard convention where $\int_a^b=-\int_b^a$ and $\int_a^b+\int_b^c=\int_a^c $ is superior, and I admit I wouldn't like to have exceptions around. How about $$\int_{a\wedge b}^{a\vee b}f(x)dx$$ instead?

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  • $\begingroup$ The standard convention really corresponds to integration of a differential form, rather than a density, or a measure. I think OP really wants good notation for integration of a density. I like his notation! $\endgroup$ Aug 28, 2014 at 20:05
  • $\begingroup$ Exact, and that's why I wouldn't suggest it. $\endgroup$ Aug 28, 2014 at 22:49
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Personally, I prefer the notation $\int_{[a,b]}f(x)dx$ which is analogous to the usual notation for higher dimensional integrals.

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