We begin by introducing the notion of a partial differential equation, as this is the topic of the book, after all.
Fix an integer \(k\ge 1\) and an open set \(U \subseteq \textbf{R}^n\). Any identity of the form \begin{equation} \tag{1} F\left(D^ku(x), D^{k-1}u(x), \dots, Du(x), u(x), x\right) = 0, \qquad \text{all }x \in \textbf{R}^d \end{equation} is called a partial differential equation (PDE) of order \(k\), where \begin{equation} F: \textbf{R}^{n^k} \times \textbf{R}^{n^{k-1}} \times \cdots \times \textbf{R} \times U \to \textbf{R} \end{equation} is given, and \(u: U \to \textbf{R}\) is the unknown (function).
To solve a PDE means to find every function \(u\) so that (1) is true. In this case, \(u\) is called a solution of the PDE. We will usually try to find only those solutions that satisfy some additional conditions on some part of the boundary of \(U\).
Fix an integer \(k\ge 1\) and an open set \(U \subseteq \textbf{R}^n\). Any identity of the form \begin{equation} \tag{1} F\left(D^ku(x), D^{k-1}u(x), \dots, Du(x), u(x), x\right) = 0, \qquad \text{all }x \in \textbf{R}^d \end{equation} is called a system partial differential equations (system of PDEs) of order \(k\), where \begin{equation} F: \textbf{R}^{mn^k} \times \textbf{R}^{mn^{k-1}} \times \cdots \times \textbf{R}^m \times U \to \textbf{R}^m \end{equation} is given, and \(u = \left(u^1, u^2, \dots, u^m\right): U \to \textbf{R}^m\) is the unknown (vector function).
A system of PDEs is essentially a collection of \(m\) interrelated PDEs for a collection of unknowns \(\left(u^1, u^2, \dots, u^m\right)\).