# Differential Operator and Trivialization Change in Lawson-Michelsohn

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!