One way to understand the symbol of a differential operator (or more generally, a pseudodifferential operator) is to see what the operator does to "wave packets" - functions that are strongly localised in both space and frequency.

Suppose, for instance, that one is working in R^n, and one takes a function psi which is localised to a small neighbourhood of a point x0, and whose Fourier transform is localised to a small neighbourhood of xi0/hbar, for some frequency xi0 (or more geometrically, think of (x0,xi0) as an element of the cotangent bundle of R^n). Such functions exist when hbar is small, e.g. psi(x) = eta( (x-x0)/eps ) e^{i xi0 . (x-x0) / hbar} for some smooth cutoff eta and some small eps (but not as small as hbar).

Now apply a differential operator L of degree d to this wave packet. When one does so (using the chain rule and product rule as appropriate), one obtains a bunch of terms with different powers of 1/hbar attached to them, with the top order term being 1/hbar^d times some quantity a(x0,xi0) times the original wave packet. This number a(x0,xi0) is the principal symbol of a at (x0,xi0). (The lower order terms are related to the lower order components of the symbol, but the precise relationship is icky.)

Basically, when viewed in a wave packet basis, (pseudo)differential operators are diagonal to top order. (This is why one has a pseudodifferential calculus.) The diagonal coefficients are essentially the principal symbol of the operator. [While on this topic: Fourier integral operators (FIO) are essentially diagonal matrices times permutation matrices in the wave packet basis, so they have a symbol as well as a relation (the canonical relation of the FIO, which happens to be a Lagrangian submanifold of phase space).]

One can construct wave packets in arbitrary smooth manifolds, basically because they look flat at small scales, and one can define the inner product xi0 . (x-x0) invariantly (up to lower order corrections) in the asymptotic limit when x is close to x0 and (x0,xi0) is in the cotangent bundle. This gives a way to define the principal symbol on manifolds, which of course agrees with the standard definition.

One way to understand the symbol of a differential operator (or more generally, a pseudodifferential operator) is to see what the operator does to "wave packets" - functions that are strongly localised in both space and frequency.

Suppose, for instance, that one is working in $\mathbb R^n$, and one takes a function $\psi$ which is localised to a small neighbourhood of a point $x_0$, and whose Fourier transform is localised to a small neighbourhood of $\xi_0/\hbar$, for some frequency $\xi_0$ (or more geometrically, think of $(x_0,\xi_0)$ as an element of the cotangent bundle of $\mathbb R^n$). Such functions exist when $\hbar$ is small, e.g. $\psi(x) = \eta( (x-x_0)/\epsilon ) e^{i \xi_0 \cdot (x-x_0) / \hbar}$ for some smooth cutoff $\eta$ and some small $\epsilon$ (but not as small as $\hbar$).

Now apply a differential operator $L$ of degree $d$ to this wave packet. When one does so (using the chain rule and product rule as appropriate), one obtains a bunch of terms with different powers of $1/\hbar$ attached to them, with the top order term being $1/\hbar^d$ times some quantity $a(x_0,\xi_0)$ times the original wave packet. This number $a(x_0,\xi_0)$ is the principal symbol of $a$ at $(x_0,\xi_0)$. (The lower order terms are related to the lower order components of the symbol, but the precise relationship is icky.)

Basically, when viewed in a wave packet basis, (pseudo)differential operators are diagonal to top order. (This is why one has a pseudodifferential calculus.) The diagonal coefficients are essentially the principal symbol of the operator. [While on this topic: Fourier integral operators (FIO) are essentially diagonal matrices times permutation matrices in the wave packet basis, so they have a symbol as well as a relation (the canonical relation of the FIO, which happens to be a Lagrangian submanifold of phase space).]

One can construct wave packets in arbitrary smooth manifolds, basically because they look flat at small scales, and one can define the inner product $\xi_0\cdot(x-x_0)$ invariantly (up to lower order corrections) in the asymptotic limit when $x$ is close to $x_0$ and $(x_0,\xi_0)$ is in the cotangent bundle. This gives a way to define the principal symbol on manifolds, which of course agrees with the standard definition.