# Homework Help: Multivariate IBP

1. Apr 5, 2010

### Kreizhn

1. The problem statement, all variables and given/known data

For $u \in \mathbb R^n$ and $P(u,y,t): \mathbb R^n \times U \times \mathbb R \to \mathbb R^n$ for some undisclosed set U, we want to evaluate

$$\int u_k \frac{\partial}{\partial u_i} \left[ u_j P(u,y,t) \right] du$$

where integration is component wise and $du = du_1 du_2 \cdots du_n$, and one is finished when all terms are expressed as

$$\int u_r P(u,y,t) du$$ for any index r.

3. The attempt at a solution

I've tried jumping straight to integration by parts, but it doesn't seem to yield anything pretty without explicitly going into cases such as "if i=j, but j $\neq$ k" yada yada. Next I tried expanding out the derivative

\begin{align*} \int u_k \frac{\partial}{\partial u_i} \left[ u_j P(u,y,t) \right] du &= \int u_k \left[ \frac{\partial u_j}{\partial u_i}P + u_j \frac{\partial P }{\partial u_i} \right] du \\ &= \int u_k \delta_{ij} P du + \int u_k u_j \frac{\partial P }{\partial u_i} du \end{align*}
Now the first term is in a state that I want it. My problem is dealing with the second term. Any ideas?