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Volume Integral symbol

  1. Sep 19, 2015 #1


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    Homework Helper

    Hi everyone.

    I've been curious about a particular symbol, but I've never seen it used or mentioned in any context. I don't really have much information about its usage, so I thought I would ask around and see if anyone knew about its application.

    Screen Shot 2015-09-19 at 3.10.24 PM.png

    I saw this symbol in Microsoft word.

    How do we interpret it, and how do we use it?

    I'm familiar with closed surface integrals with differential elements ##d \vec S##. We use those when we want to calculate the flux of a field ##\vec F##. I'm also familiar with closed surface integrals with differential elements ##dS##. We use those when we want to calculate surface area.

    What about the closed volume integral above though?

    I know we should probably use a differential element ##dV## for a closed volume, and the answer would represent the volume of the object. Is there such thing as a differential volume element ##d \vec V## such that we can extend theorems to the fourth dimension (theorem's like Stoke's theorem and the Divergence theorem)?

    Thank you in advance.
  2. jcsd
  3. Sep 19, 2015 #2


    Staff: Mentor

    I think the dash box is simply for inserting your integrand.

    Its up to you to remember the dV part.

    There is a seldom used math symbol called the delambertian thats used in relativity that is the 4D version of the del operator but this isnt it.
  4. Sep 19, 2015 #3


    Staff: Mentor

    Zondrina, I think you're asking about the integration symbol, not the box to the right. According to this page, https://en.wikipedia.org/wiki/Integral_symbol, that's a closed volume integral. I don't know much more about it, and a quick web search didn't turn up much.
  5. Sep 19, 2015 #4


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    So the only reason the loop is around the triple integral is to signify the volume is closed.

    Does that mean something like the divergence theorem can be written like so:

    Screen Shot 2015-09-19 at 6.09.47 PM.png

    For a closed volume ##V## such as ##x^2 + y^2 + z^2 \leq 1##.

    For a volume ##V## that isn't closed such as ##x^2 + y^2 + z^2 < 1##, would the theorem would take the form:

    Screen Shot 2015-09-19 at 6.12.46 PM.png

    Otherwise I don't see any reason to ever have to use the symbol mentioned in the OP.
    Last edited: Sep 19, 2015
  6. Sep 20, 2015 #5
    I don't know what is meant by "closed volume". What is the difference between a closed volume and an open volume?
  7. Sep 20, 2015 #6


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    Staff Emeritus
    Science Advisor

    In three dimensions there is no such thing as a "closed volume". There can be in higher dimensions, of course.
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