- #1

Pacopag

- 197

- 4

## 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

[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?

## 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)