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Total vs partial integration

  1. Oct 31, 2012 #1

    Is there a difference between

    [tex] \int f(x,y(x)) dx [/tex]


    [tex] \int f(x,y(x)) \partial x [/tex]


    If so, how is the total integral written in terms of partial integrals?

    Thanks for your help.
  2. jcsd
  3. Oct 31, 2012 #2


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    I have never seen [itex]\partial[/itex] used in that way.
  4. Oct 31, 2012 #3
    In both cases, the function f is dependent on x only.

    If by partial integration, you mean an iterated integral, then the result of both should be the same.

    Either way, the iterated integral uses a total differential dx, not a partial.
  5. Oct 31, 2012 #4


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    The dx usually means partial integration. The ∂x is an added reminder of partial integration it is sometimes used when solving exact differential equation as a reminder. Writing y(x) is also a clear indicator of functional dependence, more clear than writing y.
  6. Oct 31, 2012 #5


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    Neither have I, and as other posts here point out x is the only independent variable in the OP, so it cannot make any difference.
    More generally (when there's another independent variable), it could make sense as a path integral, i.e. along a path where the other independent variable is constant.
  7. Oct 31, 2012 #6


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    Have none of you read the CRC Handbook of Chemistry and Physics?
  8. Nov 1, 2012 #7


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    I've never seen ##\partial## used that way either (I haven't throughly read the CRC handbook, it seems), but if I had to wager a guess I would suppose that

    $$\int f(x,y(x))\partial x$$
    is meant to be integrated in only the first argument, holding y=y(x) fixed, while

    $$\int f(x,y(x)) dx$$
    is meant to be integrated over all of the x-dependence.

    But, without some more context, I could be entirely wrong here.
  9. Nov 1, 2012 #8
    Thanks for your help everyone. My question has been answered above and beyond.
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