 ### explicit mention of order 1 case

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 ... ... @@ -209,6 +209,25 @@ This allows us, to combine the symbols of differential operators into a space: \] \end{definition} All of this becomes a lot clearer, if one considers operators of order $1$. Let $D$ be such an operator and $g \in C^\infty_c(U)$, which we view as a multiplication operator on $C^\infty_c(M,E)$. The commutator $\benbrace*{D,g}$ acts on $u \in C^\infty_c(U,E)$ via $\enbrace[\big]{\benbrace*{D,g}u}(x) = \sum_{i=1}^{d} a_i(x) \diff{g}{x_i}(x) u(x)$ In other words $\benbrace*{D,g}$ is acting via the following element of $C_c^\infty(U,\End(E))$ $x \mapsto \sum_{i=1}^{d} a_i(x) \diff{g}{x_i}(x)$ The computation in the proof of \cref{lem:symbol_welldefined} now shows: \begin{lemma} If $D$ is an operator of order $1$, the symbol can be computed as $\sigma_1(D)(\mathd g) u = i \,\benbrace*{D,g} u$ \end{lemma} The symbol gives raise to an extremly important class of differential operators: \begin{definition} ... ...
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