Given a nonnegative integer
m, an order-m linear differential operator is a map P from a
function space \mathcal{F}_1 on \mathbb{R}^n to another function space \mathcal{F}_2 that can be written as: P = \sum_}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots\partial x_n^{\alpha_n}} Thus for a function f \in \mathcal{F}_1: P f = \sum_f}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots\partial x_n^{\alpha_n}} The notation D^{\alpha} is justified (i.e., independent of order of differentiation) because of the
symmetry of second derivatives. The polynomial
p obtained by replacing partials \frac{\partial}{\partial x_i} by variables \xi_i in
P is called the
total symbol of
P; i.e., the total symbol of
P above is: p(x, \xi) = \sum_{|\alpha|\le m}a_\alpha(x) \xi^\alpha where \xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n}. The highest homogeneous component of the symbol, namely, :\sigma(x, \xi) = \sum_{|\alpha|= m}a_\alpha(x) \xi^\alpha is called the
principal symbol of
P. While the total symbol is not intrinsically defined, the principal symbol is intrinsically defined (i.e., it is a function on the cotangent bundle). More generally, let
E and
F be
vector bundles over a manifold
X. Then the linear operator : P: C^\infty(E) \to C^\infty(F) is a differential operator of order k if, in
local coordinates on
X, we have : Pu(x) = \sum_{|\alpha| = k} P^\alpha(x) \frac {\partial^\alpha u} {\partial x^{\alpha}} + \text{lower-order terms} where, for each
multi-index α, P^\alpha(x):E \to F is a
bundle map, symmetric on the indices α. The
kth order coefficients of
P transform as a
symmetric tensor : \sigma_P: S^k (T^*X) \otimes E \to F whose domain is the
tensor product of the
kth
symmetric power of the
cotangent bundle of
X with
E, and whose codomain is
F. This symmetric tensor is known as the
principal symbol (or just the
symbol) of
P. The coordinate system
xi permits a local trivialization of the cotangent bundle by the coordinate differentials d
xi, which determine fiber coordinates ξ
i. In terms of a basis of frames
eμ,
fν of
E and
F, respectively, the differential operator
P decomposes into components :(Pu)_\nu = \sum_\mu P_{\nu\mu}u_\mu on each section
u of
E. Here
Pνμ is the scalar differential operator defined by :P_{\nu\mu} = \sum_{\alpha} P_{\nu\mu}^\alpha\frac{\partial}{\partial x^\alpha}. With this trivialization, the principal symbol can now be written :(\sigma_P(\xi)u)_\nu = \sum_{|\alpha|=k} \sum_{\mu}P_{\nu\mu}^\alpha(x)\xi_\alpha u_\mu. In the cotangent space over a fixed point
x of
X, the symbol \sigma_P defines a
homogeneous polynomial of degree
k in T^*_x X with values in \operatorname{Hom}(E_x, F_x) . == Fourier interpretation ==