> [!definition] > > Let $U \subset \real^d$ be [[Open Set|open]], then a **linear partial differential operator** is a linear map of the form > $ > L = \sum_{\abs{\alpha} \le m}a_\alpha D^\alpha > $ > where each $a_\alpha: U \to \complex$ is a coefficient function, with varying regularity depending on the space that $L$ acts on. The smallest $m \in \nat$ such that $L$ can be written as the above sum is the **order** of $L$. If each $a_\alpha$ is constant, then $L$ has **constant coefficients**. > > The mapping $P: U \to \mathbb \complex [\xi_1, \cdots, \xi_d]$ defined by $\xi \mapsto \sum_{\abs{\alpha} \le m}a_\alpha \xi^\alpha$ such that $L = P(D)$ is the **symbol** of $L$. If $P_m \ne 0$, then $P_m$ is the **principal symbol** of $L$.