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Multivariate IBP

  1. Apr 5, 2010 #1
    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
    Now the first term is in a state that I want it. My problem is dealing with the second term. Any ideas?
  2. jcsd
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