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I am currently reading the excelent book by Lawson and Michelsohn on Spin Geometry. In chapter 3 paragraph 1, they define a differntial operator of order $m$ to be a $\mathbb{C}$-linear map of $\mathbb{C}$-vector bundles of ranks $p$ respectively $q$, as $P\colon E\longrightarrow F$, in the local coordinate $(U,x)$ and local trivializations $\Phi_E\colon \left.E\right|_U\longrightarrow U\times\mathbb{C}^p,\Phi_F\colon \left.F\right|_U\longrightarrow U\times\mathbb{C}^q$ by $P=\sum_{\left|\alpha\right|\leq m}A^\alpha(x)\frac{\partial^{\left|\alpha\right|}}{\partial x^\alpha}$.

When applying a change of trivialization, say $g_E\colon U\cap V\longrightarrow \operatorname{GL}_p(\mathbb{C})$ and $g_F\colon U\cap V\longrightarrow \operatorname{GL}_p(\mathbb{C})$ they then claim that in the new trivializations $\Psi_E\colon \left.E\right|_V\longrightarrow V\times\mathbb{C}^p,\Psi_F\colon \left.F\right|_V\longrightarrow V\times\mathbb{C}^q$ the operator $P$ is of the form $P=g_F(\sum_{\left|\alpha\right|\leq m}A^\alpha(x)\frac{\partial^{\left|\alpha\right|}}{\partial x^\alpha})g_E^{-1}=\sum_{\left|\alpha\right|\leq m}\tilde A^\alpha(x)\frac{\partial^{\left|\alpha\right|}}{\partial x^\alpha}$ where $\tilde A^\alpha=g_F A^\alpha g_E^{-1}$ whenever $\left|\alpha\right|=m$. I have tried to do this computation for some time now but I cannot seem to do it. If $\sigma\in\Gamma(E)$ is given under the identification $\Gamma(\left.E\right|_U)\leftrightarrow (C^\infty(M,\mathbb{C}))^p$ by $\sigma_\Phi=(\sigma_\Phi^1,\dots,\sigma_\Phi^p)$ in the trivialization $\Phi_E$ and by $\sigma_\Psi=(\sigma_\Psi^1,\dots,\sigma_\Psi^p)$ in $\Psi_E$, then $\sigma_\psi^j=\sum_{i=1}^p (g_E^{-1})_{ij}\sigma_\Phi^i$. So by the generalised Leibniz' rule we find $$ \frac{\partial^{\left|\alpha\right|}}{\partial x^\alpha}\sum_{i=1}^p (g_E^{-1})_{ij}\sigma_\phi^ i=\sum_{\nu\leq\alpha}\begin{pmatrix}\alpha \\ \nu\end{pmatrix}\sum_{i=1}^p\frac{\partial^{\left|\nu\right|}}{\partial x^\nu}(g_E^{-1})_{ij}\frac{\partial^{\left|\alpha-\nu\right|}}{\partial x^{\alpha-\nu}}\sigma_\Phi^i, $$ but I don't think that helps much. If anyone knows how to do this computation, I would be eager to know!

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