In Chapter 6 of Warner's Foundations of Differentiable Manifolds and Lie Groups, he develops a self-contained theory of local elliptic operators to establish the Hodge theorem.
I got a bit stuck on a criterion that can examine ellipticity of a linear differential operators in a coordinate-free way. For reference, let me recall some definitions, which are taken nearly verbatim from Warner's book.
Let $L$ be a linear differential operator $L$ of order $l$ on the space of all smooth maps from $\mathbb{R}^n$ to $\mathbb{C}^m$. Write $L=P_{l}\left(D\right)+\cdots +P_0\left(D\right)$, where $P_j\left(D\right)$ is a $m\times m$ matrix whose entries are of the form $$ \sum_{\left[\alpha\right]=j}a^{\alpha}D^{\alpha} $$ in which $\alpha=\left(\alpha_1,\cdots,\alpha_n\right)$ is a multi-index, $\left[\alpha\right]=\sum_{i=1}^{n}\alpha_{i}$, $D^{\alpha}=\left(\frac{1}{\sqrt{-1}}\right)^{\left[\alpha\right]}\frac{\partial^{\left[\alpha\right]}}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}$, and $a^{\alpha}:\mathbb{R}^n\to \mathbb{C}$ is a smooth function for each $\alpha$.
Let $\xi\in \mathbb{R}^n$ and let $P_j\left(\xi\right)$ denote the $m\times m$ matrix whose entries are obtained by substituting $\xi^{\alpha}$ for $D^{\alpha}$. Here, $\xi^{\alpha}=\xi_{1}^{\alpha_{1}}\cdots \xi_{n}^{\alpha_{n}}$.
Definition. $L$ is say to be elliptic at $x\in \mathbb{R}^n$ if the matrix $P_{l}\left(\xi\right)$ is non-singular at $x$ for each non-zero $\xi$.
The Criterion.
$L$ is elliptic at $x$ if and only if $$ L\left(\varphi^{l}u\right)\left(x\right)\neq 0 $$ for each $\mathbb{C}^m$-valued function $u$ such that $u\left(x\right)\neq 0$ and each smooth real-valued function $\varphi$ such that $\varphi\left(x\right)=0$ and $d \varphi \left(x\right)\neq 0 \in T^{*}_{x}\mathbb{R}^n\simeq \mathbb{R}^n$.
Question. Warner suggests that it follows from the equation $$ L\left(\varphi^{l}u\right)=P_{l}\left(d\varphi|_{x}\right)\left(u\left(x\right)\right), \tag{1} $$ yet it does not seem right to me. I guess the R.H.S. of (1) should be $$ l!P_{l}\left(d\varphi|_{x}\right)\left(u\left(x\right)\right) $$ since, by the chain rule, the (partial) derivative of $\varphi^{l}$ provides its power. Actually, I do try some specific example to verify my hunch as follows.
Let $L= \begin{bmatrix} \frac{\partial^2}{\partial x^2} &0\\ 0&\frac{\partial }{\partial x}\frac{\partial }{\partial y}\\ \end{bmatrix}$ on the space of all smooth maps from $\mathbb{R}^2$ to $\mathbb{C}^2$. Then $$ L\left(\varphi^2u\right)=\begin{bmatrix} 2\left(\frac{\partial \varphi}{\partial x}\right)^2 u_{1}\left(x\right) \\ 2\frac{\partial \varphi}{\partial x}\frac{\partial \varphi}{\partial y} u_{2}\left(x\right)\\ \end{bmatrix}=2P_{l}\left(d\varphi|_{x}\right) \left( u \left( x\right) \right) $$ for any $\varphi$ and $u=\left(u_1,u_2\right)$ that satisfy the conditions in the criterion. This suggests we need some constant included in (1).
Do I miss something or misunderstand the criterion? Any help or comment is absolutely appreciated!
Remark. I still want to know if my point is right, although it does not really affect the invertibility of the matrix $P_{l}\left(\xi\right)$ regardless of the presence of the constant.