Differential operators and principal symbols. For vector bundles $E$ and $F$ on a manifold $M$ I have seen here that a linear differential operator $L:\Gamma(E)\to\Gamma(F)$ of order $k$ can locally be written in the form $L=\sum_{\alpha\leq k}l_{\alpha}\partial^{\alpha}$, $l_{\alpha}:E\to F$ is a bundle homomorphism.


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*How do I calculate $\partial^{\alpha}S$ when $S\in\Gamma(E)$? Do I simply pick a basis/frame for $E$ and apply $\partial^{\alpha}$ to the component functions?


The symbol of $L$ in the direction of a one-form $\omega$ is the differential operator obtained by only summing over those $\alpha$ for which $\lvert\alpha\rvert=k$ and replacing $\partial^{\alpha}$ by $\omega^{\alpha}=\omega_1^{\alpha_1}\dots\omega_n^{\alpha_n}$, with $(\omega_i)_{i=1}^n$ being the components of $\omega$ in dual basis/frame. In other words, $\text{Principal symbol}(L)(\omega)=\sum_{\lvert\alpha\rvert=k}\omega^{\alpha}l_{\alpha}.$ In a calculation from this answer here, covariant derivatives, rather than derivatives, were replaced with the components of the one-form (in this one they use $\xi$ instead of $\omega$).


*How does the definition of the principle symbol coincide with the answer posted there?

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*How do I calculate $\partial^\alpha S$ when $S \in \Gamma(E)$? Do I simply pick a basis/frame for E and apply $\partial^\alpha$ to the component functions?
  

Yes, that is exactly what you do. Implicit in what you wrote is that one has made choices of a local trivialization of $E$ and of local coordinates on $M$. The decomposition of $L$ into terms of the form $l_\alpha \partial^\alpha$ depends on those choices.
To be more explicit: Let $U \subset M$ be an coordinate open set with coordinates $\{x^a\}$ (more generally, we could just choose a local frame for $TM$ that need not be a coordinate frame) on which $E$ is trivial, so there exists a local frame $\{e^i\}$. We may write a local section $S \in \Gamma(E|_U)$ in terms of the local frame $\{e^i\}$ as $S = \sum_i f_i e^i$, where each $f_i$ is a scalar-valued function on $U$. Then
$$ (l_\alpha \partial^\alpha) (S) =  \sum_i \partial^\alpha( f_i) l_\alpha (e^i) ,$$
which is a local section of $F$ since $l_\alpha$ is a local bundle homomorphism $E \to F$, meaning $l_\alpha(e^i)$ is a local section of $F$.


  
*How does the definition of the principal symbol coincide with the answer posted there?
  

The two definitions are equivalent. With respect to a local frame for $E$, a connection $\nabla$ is just the exterior derivative $d$ (operating on the coefficient functions, as above) plus a zero-order term (sometimes called the "connection one-form"). Another way of saying this is that $\nabla_a = \frac{\partial}{\partial x^a} + \Gamma_a$, where $\Gamma_a$ is a local section of $\text{End}(E)$. This means that replacing $\frac{\partial}{\partial x^a}$ by $\nabla_a$ does not affect the leading order (order $k$ according to your notation) terms, which determine the principal symbol.
