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Integration by parts (multivariable)

  1. Mar 1, 2008 #1
    1. The problem statement, all variables and given/known data
    I can find on Wikipedia the "formula" for integration by parts for the case where there is a multi-variable integrand, but I would like to know what substitutions to make in order to show my steps.


    2. Relevant equations
    For multiple variables we have
    [tex]\int_{\Omega}{{\partial u}\over{\partial x_i}}vdx=-\int_{\Omega}{{\partial v}\over{\partial x_i}}udx[/tex].
    assuming that we can drop the surface term for physical reasons. Here, u and v are functions of several variables, say {[tex]x_1, x_2, ....x_n[/tex]}
    First of all, should the [tex]dx[/tex] be a [tex]dx_i[/tex] ??
    Now, my real question is; what substitutions do I make in order to show this?


    3. The attempt at a solution
    I feel like the generalization from 1D to the above higher-D version should be obvious, but it just isn't to me. I guess what is bothering me is that e.g.
    [tex]du = \sum {{\partial u}\over{\partial x_i}}dx_i[/tex]. And I can't get this to fit into a derivation of the equation in 2. (above)
     
  2. jcsd
  3. Mar 1, 2008 #2

    HallsofIvy

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    Just as in one variable calculus, what you choose to be u and what you choose to be dv depends heavily on the particular function you are trying to integrate! Do you have a particular example to show us?
     
  4. Mar 1, 2008 #3
    I don't have particular functions to work with. I have a sum like
    [tex]\sum_{i} \int dx_{1}...dx_{n}dp_{1}...dp_{n} A t \left( {\partial H \over {\partial x_{i}}}{\partial \rho \over \partial p_{i}}-{\partial H \over {\partial p_{i}}}{\partial \rho \over \partial x_{i}} \right)[/tex],
    and I am just trying to bring the derivatives from [tex]\rho[/tex] over to the A. Here, [tex]\rho[/tex] and A are functions of the x_i and p_i's, and t is time, i.e an independent variable.

    What would be nice is a derivation of the integration by parts formula in part 2. of my original post
     
    Last edited: Mar 1, 2008
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