(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

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

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

[tex] \int u_r P(u,y,t) du [/tex] 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 [itex] \neq [/itex] k" yada yada. Next I tried expanding out the derivative

[tex] \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*}

[/tex]

Now the first term is in a state that I want it. My problem is dealing with the second term. Any ideas?

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# Homework Help: Multivariate IBP

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