# Simplification Trick

1. Apr 2, 2010

### Kreizhn

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

If it turns out that
$$\frac{du_i}{dt} = \sum_{j=1}^N A_{ij}(y)u_j$$
this is supposed to simplify $\frac{\partial P}{\partial t}$ 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

$$- \sum_{i=1}^N \left[\frac{\partial}{\partial u_i} F_i(u,t,y) \right] P$$
In this case this is equivalent to the divergence of a linear vector field, and if we define A such that $(A)_{ij} = A_{ij}$ then indeed this does simplify to
$$\frac{\partial P}{\partial t} = - \text{Tr}[A] P$$
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.