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Homework Help: Simplification Trick

  1. Apr 2, 2010 #1
    1. The problem statement, all variables and given/known data
    I'm given a differential equation that evolves as
    [tex] \frac{du}{dt} = F(u,t; y(t) ), \qquad u \in \mathbb R^n [/tex]
    and told that a vector valued function P(u,t,y) satisfies
    [tex] \frac{\partial P}{\partial t} = - \sum_{i=1}^N \frac{\partial}{\partial u_i} F_i(u,t,y) P [/tex]

    If it turns out that
    [tex] \frac{du_i}{dt} = \sum_{j=1}^N A_{ij}(y)u_j [/tex]
    this is supposed to simplify [itex] \frac{\partial P}{\partial t} [/itex] greatly.

    3. The attempt at a solution

    I don't see how this simplifies at all. One possibility is that the question is poorly written and the summation term should actually be

    [tex] - \sum_{i=1}^N \left[\frac{\partial}{\partial u_i} F_i(u,t,y) \right] P [/tex]
    In this case this is equivalent to the divergence of a linear vector field, and if we define A such that [itex] (A)_{ij} = A_{ij} [/itex] then indeed this does simplify to
    [tex] \frac{\partial P}{\partial t} = - \text{Tr}[A] P [/tex]
    However, I have reason to believe that the equation is not wrongly written, in which case substituting our value for F into the differential equation for P doesn't drammatically simply the problem as suggested.
     
  2. jcsd
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