Suppose $P=\sum_i a_i \partial/\partial x^i$. It seems for me that there always exists a nonzero $\xi=\{\xi_1, \cdots, \xi_n\}\in \mathbb R^n\setminus 0$ such that the principle symbol $\sigma(P)(x, \xi)=\sum_i a_i\xi_i=0$. Hence by definition it cannot be elliptic. Did I miss anything?
So for differential operators acting on functions defined over $U\subset\mathbb R^n$, The only elliptic operator must be of second order?
How to define the symbol of $P$ for function $f:\mathbb R^n\to\mathbb R^n$? Can you show me an example of first order ellliptic operator?