# Examples for differential operators first order

Currently I am dealing with the problems which involve the general first order differential operator, i.e., for some open domain $\Omega\subset\mathbb{R}^n$ with certain regular bondary and a function $u:\Omega\rightarrow \mathbb{R}^l$ with at least one times differentiability, we define its general derivative by the vector field $$(\sum_{ij}a_{ijk}\partial_i u_j)_k\in\mathbb{R}^m,$$

where $a_{ijk}$ are constants. Now I have some examples which appear mostly in practice, they are gradient, divergence and curl operators. But most examples that I have seen are more or less related the the previous three types. I should give some general theory if the constants $a_{ijk}$ satisfy some certain conditions. But I am not familiar in mechanics or electronics, I don't know if the there are also other types of derivatives exist in practice which are diffeerent from the previous three types. For instance, I can define the derivatives for $n=2,m=l=1$ by $$\operatorname{div}u\begin{pmatrix}1\\1\end{pmatrix} +\begin{pmatrix}\partial_1 u_2\\\partial_2 u_1\end{pmatrix}$$ or, let $l=n,m=1$ and define the derivative by $\partial_1 u$. I can in fact give many examples, but if they make sense, I am not so sure. Also, a general theory is hard to give if the constants have too less restrictions. So I need to know if there are some other examples that also more or less appear in parctice or in references.

Thank you for any kind hints!

• A fairly well studied class of systems that fit your criteria are "symmetric hyperbolic" equations, which is a useful search term. – Igor Khavkine Jul 31 '15 at 10:19
• Also, here's a paper that discusses such systems that are of elliptic type: Hile & Protter, Properties of overdetermined first order elliptic systems (1977). – Igor Khavkine Jul 31 '15 at 10:23
• @IgorKhavkine Thanks for your hint! – Peter Jul 31 '15 at 12:03