Let \(E\) and \(F\) be complex \(C^{\infty}\) vector bundles over a differentiable manifold \(X\). Let \(\underline{C}^{\infty}(E)\) and \(\underline{C}^{\infty}(F)\) be the corresponding sheaves of smooth sections, and let \(P\colon\underline{C}^{\infty}(E)\to\underline{C}^{\infty}(F)\) be a \(\mathbb{C}\)-linear morphism of sheaves. This discussion works the same over \(\mathbb{R}\) instead of \(\mathbb{C}\) as well.

We say that \(P\) is a differential operator of order \(k\) if, in common trivializations over open sets \(U\subseteq X\) with coordinates \(x_1,x_2,\dots,x_n\) where \[E|_U\cong U\times\mathbb{C}^p,\qquad F|_U\cong U\times\mathbb{C}^q,\] we have \(P((\alpha_1,\alpha_2,\dots,\alpha_p))=(\beta_1,\beta_2,\dots,\beta_q)\) with \[\beta_i=\sum_{I,j}{P_{I,i,j}\frac{\partial\alpha_j}{\partial x_I}}\] where the coefficients \(P_{I,i,j}\) are \(C^{\infty}\), and zero for \(|I|>k\), with at least one coefficient \(P_{I,i,j}\) nonzero for \(|I|=k\).

Let us take a look at the order \(k\) part of \(P\) in the trivialization coordinates. This operator \(P^k\) can be written as a matrix \[[P^k_{i,j}]=\sum_{|I|=k}{P_{I,i,j}\frac{\partial}{\partial x_I}}.\] We claim that the factors \(\frac{\partial}{\partial x_I}\) transform like sections of the \(k\)th symmetric power bundle \(S^k(T(X))\) under changes of coordinates on \(X\). Indeed, let \(U\) be an open subset on which both \(E\) and \(F\) are trivial, and let \(x_1,x_2,\dots,x_n\) and \(y_1,y_2,\dots,y_n\) be coordinate systems on \(U\). Let \(\Phi\colon U\to U\) be the change of coordinates with \(\Phi(x_i)=y_i\). Observe that by the chain rule, \[\frac{\partial}{\partial x_i}=\sum_{j}{\frac{\partial\Phi_j}{\partial x_i}\frac{\partial}{\partial y_j}},\] hence as sections of \(S^k(T(X))\) we have \[\prod_{\ell=1}^{k}{\frac{\partial}{\partial x_{i_{\ell}}}}=\prod_{\ell=1}^{k}{\sum_{j=1}^{n}{\frac{\partial\Phi_j}{\partial x_{i_{\ell}}}\frac{\partial}{\partial y_j}}}.\] We can verify that the right hand side of the above equation is also equivalent to \(\frac{\partial}{\partial x_{i_1}\partial x_{i_2}\dots\partial x_{i_k}}\) by induction on \(k\). We have already established the base case of \(k=1\). Suppose the claim is true for some \(k\). Put \(S_{\ell}=\sum_{j=1}^{n}{\frac{\partial\Phi_j}{\partial x_{i_{\ell}}}\frac{\partial}{\partial y_j}}\). Then by the Leibniz rule, we have \[\frac{\partial}{\partial x_{i_1}\partial x_{i_2}\dots\partial x_{i_{k+1}}}=\sum_{m=1}^{k}{\frac{\partial S_m}{\partial x_{i_{k+1}}}\prod_{m\neq \ell=1}^{k}{S_{\ell}}}.\] By chain rule, we have \[\frac{\partial S_m}{\partial x_{i_{k+1}}}=\sum_{t=1}^{n}{\frac{\partial\Phi_t}{\partial x_{i_m}}\sum_{s=1}^{n}{\frac{\partial\Phi_s}{\partial x_{i_{k+1}}}\frac{\partial}{\partial y_s\partial y_j}}}.\] Combining this with the above, a moment’s thought reveals that we will have completed the induction.

A similar calculation will show that the matrix of coefficients \([P_{I,i,j}]\) describes a morphism of bundles \(E|_U\to F|_U\) and these matrices transform in the same way as a section of \(\mathrm{Hom}{(E,F)}\) upon changes of trivialization. Therefore, the \(k\)th order data of \(P\) is captured by a section \(\sigma_P\) of the bundle \(\mathrm{Hom}{(E,F)}\otimes S^k(T(X))\). This section \(\sigma_P\) is known as the symbol of the operator \(P\).

Now we may recall two facts from linear algebra.

  1. For vector spaces \(V\) and \(W\), if \(V\) is finite dimensional, then \(\mathrm{Hom}{(V,W)}\cong V^*\otimes W\).
  2. For a vector space \(V\) over a field of characteristic 0, we have that \(S^k(V^*)\cong S^k(V)^*\). See here and here as references.

From these facts, it follows that \[\mathrm{Hom}{(E,F)}\otimes S^k(T(X))\cong\mathrm{Hom}{(S^k(T^*(X)),\mathrm{Hom}{(E,F)})}.\] The symbol \(\sigma_P\) is often interpreted as a section of the bundle on the right hand side. This means that at each point \(x\in X\), we can interpret \(\sigma_P\) as giving homogeneous map of degree \(k\) from the cotangent space to \(\mathrm{Hom}{(E_x,F_x)}\). If \(\sigma_{P,x}(\alpha_x)\colon E_x\to F_x\) is injective for each \(x\) and nonzero covector \(\alpha_x\), then we say that \(P\) is elliptic.