I want to prove the principle symbol is globally defined as an element \begin{align} \Gamma(T^* M / \{0\}, \textrm{Hom}(\pi^*E, \pi^* F) ) \end{align}
To more specify, let me explain the definition of principle symbol which i mention above.
The principle symbol of $L$ is defined by \begin{align} \sigma_{k}(L)(x,\xi) = \sum_{j=1, |\alpha|=k}^p a_{\alpha}^{ij} (x) \xi^\alpha \end{align} where $\xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n}$.
I want to prove this symbol is globally defined.
Any suggestion?
This is an extension of my comment rather than a complete answer, but as indicated, this is a bit involved.. The basic idea of jet bundles is that to a vector bundle $E\to M$ and an integer $k$, one can naturally associate a vector bundle $J^kE\to M$ called the $k$-th jet prolongation. An element of $J^kE$ over a point $x\in M$ is an equivalence class of smooth sections of $E$ defined on some open neighbourhood of $x$, with two sections being equivalent if and only if they have the same value and the same Taylor-development up to order $k$ in $x$ in local charts. Local charts for $M$ together with local trivializations of $E$ can be used to construct local charts for $J^kE$ (in which the Taylor-coefficients are coordinates).
A linear differential operator $\Gamma(E)\to\Gamma(F)$ of order at most $k$ is then equivalent to a vector bundle map $J^kE\to F$. (Basically, there is an obvious operator $\Gamma(E)\to\Gamma(J^k(E))$ which maps a section $s$ to its equivalence classes in each point. This is a linear differential operator of order $k$ and then you just compost with the bundle map.)
Now there is an obvious vector bundle map from $J^kE$ onto $J^{k-1}E$ (which just corresponds to a coarser equivalence relation). The kernel of this projection is naturally isomorphic to $S^kT^*M\otimes E$, whose fiber in a point is the space of $k$-linear symmetric maps $(T_xM)^k\to E_x$. (This essentially is due to the fact that for a function vanishing to $(k-1)$st order in a point, the $k$th dervative is largely independent of choices of coordinates).
Given a differential operator $D:\Gamma(E)\to\Gamma(F)$ of order $\leq k$, consider the corresponding bundle map $\tilde D:J^kE\to F$ and restrict it to the kernel of the projection to $J^{k-1}E$. What you get is a well defined bundle map $S^kT^*M\otimes E\to F$, or equivalently $S^kT^*M\to L(E,F)$, which is the principal symbol of $D$. The form you use in the question is obtained from this one via interpreting this symbol as a Section of $S^kTM\otimes L(E,F)$ and then viewing elements of $S^kTM$ as homomogeneous polynomials of order $k$ on the fibers of $T^*M$.