Integration by parts (multivariable)

[Pacopag]

Homework Statement

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.

Homework Equations

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

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.
$$du = \sum {{\partial u}\over{\partial x_i}}dx_i$$. And I can't get this to fit into a derivation of the equation in 2. (above)

 

[HallsofIvy]

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?

 

[Pacopag]

I don't have particular functions to work with. I have a sum like
$$\sum_{i} \int dx_{1}...dx_{n}dp_{1}...dp_{n} A t \left( {\partial H \over {\partial x_{i}}}{\partial \rho \over \partial p_{i}}-{\partial H \over {\partial p_{i}}}{\partial \rho \over \partial x_{i}} \right)$$,
and I am just trying to bring the derivatives from $$\rho$$ over to the A. Here, $$\rho$$ 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