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I can't find this online from a simple search, and I cannot remember.

Given two curves/path $C$ and $D$, what is the notation for path concatenation when describing a path integral? Here are some ideas I came up with. Is there a canonical notation?

  • $\int_{CD} f \cdot d\vec r$
  • $\int_{C+D} f \cdot d\vec r$
  • $\int_{C \circ D} f \cdot d\vec r$
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Since they're conceived of as sets of points, the "concatenation" would presumably be their set union $C\cup D$... –  anon Jul 29 '11 at 17:18
Makes sense. If you post that as an answer, I will accept it. –  Michael Chen Sep 4 '11 at 15:13
I don't think there's a canonical notation. However, the notation $C + D$ makes a lot of sense in the context of singular homology. –  Jesse Madnick Sep 4 '11 at 19:05
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1 Answer

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The paths are each considered sets of points, so concatenating them (gluing them together or using them as separate components both in the integration) involves taking all of the points of the paths jointly, which would be the set union $C\cup D$. I believe I've seen this notation in integrals but I can't remember a specific source off the top of my head. Note that if $C$ and $D$ share a measurable piece of curve together (or space generally for multivariable integrals), $\oint_{C\cup D}$ will only integrate over it once instead of twice like $\oint_C+\oint_D$ would.

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