> [!definition] > > Let $d \in \real^d$, $\Omega \subset \real^d$, and $L = \sum_{\alpha}a_\alpha D^\alpha$ be a [[Linear Partial Differential Operator|linear partial differential operator]] on $\Omega$ of order $k$, then the homogeneous polynomial > $ > \chi_{L}(x, \xi) = \text{Char}_x(L) = \sum_{|\alpha| = k}a_\alpha(x) \xi^\alpha > $ > corresponding to the top order terms. Let $x \in \real^d$ and $v \in \real^d$ with $v \ne 0$, then $v$ is **characteristic** for $L$ **at** $x$ if $\chi_L(x, v) = 0$. The zero set $V(\chi_L(x, \cdot))$ is the **characteristic variety** of $L$ at $x$. > [!theorem] > > Let $F: \Omega \to \Omega'$ be a [$C^\infty$](Space%20of%20Continuously%20Differentiable%20Functions)-isomorphism. If $L$ has $C^\infty$ coefficients, then $F$ induces a partial differential operator > $ > L': C^\infty(\Omega') \to C^\infty(\Omega') \quad f \mapsto L(f \circ F) \circ F^{-1} > $ > where > $ > \chi_{L'}(y, \xi) = \chi_L(F^{-1}(y), DF_{F^{-1}(y)}^*(\xi)) > $ > > *Proof*. Let $x \in X$, then via [[Pushforward of Vector Field|pushforwards]], > $ > \begin{align*} > (L'f)(y) &= L(f \circ F)(F^{-1}(y)) \\ > &= \sum_{\abs{\alpha} \le k}a_\alpha(F^{-1}(y))[(DF_{F^{-1}(y)}^*\partial_y)^\alpha f](F^{-1}(y)) > \end{align*} > $ > which has top order terms $\sum_{\abs{\alpha} = k}DF_{F^{-1}(y)}^*\partial_y^\alpha$. Therefore > $ > \chi_{L'}(y, \xi) = \sum_{\abs{\alpha} = k}a_\alpha(F^{-1}(y))(DF_{F^{-1}(y)}^* \xi)^\alpha > $ > [!theorem] > > Let $S \subset \real^d$ be a [[Hypersurface|hypersurface]] of class $C^1$ and $\nu: S \to \real^d$ be a normal vector field, then $S$ is **characteristic** for $L$ **at** $x \in S$ if $\chi_L(x, \nu(x)) = 0$, and $S$ is **non-characteristic** for $L$ if $S$ is not characteristic for $L$ at all $x \in S$. > > Let $\Omega \supset S$ be [[Open Set|open]], $F: \Omega \to \Omega'$ be a $C^\infty$-isomorphism, then a normal vector field of $F(S)$ is defined by > $ > F(\nu):S \to \real^d \quad y \mapsto (DF_{F^{-1}(y)}^*)^{-1}(\nu(F^{-1}(y))) > $ > Therefore $\chi_{L'}(y, F(\nu)(y)) = 0$ if and only if $\chi_L(x, \nu(x)) = 0$. Hence the characteristic property is invariant under diffeomorphisms.