1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Science Advisor

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook