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I Shape of infinitesimal volumes

  1. Dec 2, 2017 #1
    In physics we often use objects with infinitesimal volume. An example is the infinitesimal volumes that we use to calculate the electrostatic field knowing the charge distribution.
    Very often in the books i studied these infinitesimal elements are represented as infinitesimal cubes.
    My question: can i utilize infinitesimal elements of an arbitrary shape ? For example spherical ? And why is it possible?
  2. jcsd
  3. Dec 2, 2017 #2


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    It depends on what coordinates you are using. With plane polar coordinates, for example, an area element has the form:

    ##dr \times rd\phi##

    and is a curvilinear wedge shape.
  4. Dec 2, 2017 #3


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    Yes, as long as you avoid doing anything too terribly pathological. For example, the general expression for the mass within a volume is ##\int_V\mu(....)~\mathrm{d}V## where ##\mu## is the density as a function of position. If the mass distribution is spherically symmetric then we can use spherical coordinates and the most convenient volume element is a spherical shell: ##\mathrm{d}V=4\pi{r}^{2}\mathrm{d}r##, and the integral is ##\int_0^\infty{4}\pi\mu(r){r}^{2}~\mathrm{d}r##. The choice is mostly one of convenience - in this case you wouldn't want to use a cubical volume element (although a masochist could, after writing ##\mu## as a function of ##x##, ##y##, and ##z##).

    As for why it works.... It's basically the same limiting process as used for all integrals.
    Last edited: Dec 2, 2017
  5. Dec 2, 2017 #4


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    You can use any shape, but some shapes will require much more effort to do the algebra than others without offering any benefit for the choice.
    We typically make the choice that makes the work easiest.
  6. Dec 3, 2017 #5


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    That's my general approach to life! :smile:
  7. Dec 3, 2017 #6


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    The small shapes must be capable of being stacked together to form a solid large shape with no gaps between the pieces.

    Small rectangular blocks can be stacked without gaps.

    Nugatory in post #3 gave an example of concentric hollow spherical shells that can be stacked inside each other with no gaps.

    On the other hand, small solid spherical balls can't be stacked side-by-side without gaps, so you can't use those.
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