(adsbygoogle = window.adsbygoogle || []).push({}); 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'tdrammaticallysimply the problem as suggested.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Simplification Trick

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**