Linear system in polar coordinates

[Jhenrique]

Hellow! I have searched for some theory about linear system in polar coordinates, unfortunately, I not found anything... exist some theory, some book, anything about this topic for study? Thanks!

 

[Stephen Tashi]

Give an example of what you mean by "linear system in polar coordinates". Are you talking about a set of simultaneous linear equations? - or a system of linear differential equations?

 

[Jhenrique]

I think in something like:
$$\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix} \begin{bmatrix} r\\ \theta\\ \end{bmatrix} = \begin{bmatrix} b_{1}\\ b_{2}\\ \end{bmatrix}$$
Where a_ij can be a simple coeficient or a polynomial of kind aD³+bD²+cD+d (where D is the derivative operator) and b_i a simple coeficient.

But using [r θ] instead of use [x y].

 

[Stephen Tashi]

Where a_ij can be a simple coeficient or a polynomial of kind aD³+bD²+cD+d (where D is the derivative operator) and b_i a simple coeficient.

In that equation, I don't understand what function D would operate upon? Do we have funtions $r = r(t), \theta = \theta(t)$ and $D = \frac{d}{dt}$ ?