Multivariate IBP Homework: Evaluating ∫ukduiP(u,y,t) du

[Kreizhn]

Homework Statement

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.

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?

 

[mighty2000]

One approach you could try is using the product rule for derivatives to expand out the second term:

\begin{align*}
\int u_k u_j \frac{\partial P}{\partial u_i} du &= \int u_k \left( u_j \frac{\partial P}{\partial u_i} + P \frac{\partial u_j}{\partial u_i} \right) du \\
&= \int u_k u_j \frac{\partial P}{\partial u_i} du + \int u_k P \frac{\partial u_j}{\partial u_i} du
\end{align*}

Now, the first term on the right-hand side is the same as the original integral, so you can substitute it in and simplify. For the second term, you can use the chain rule to express it in terms of the original function P. This should help you to simplify the integral and express it in the desired form.