Principal symbol

The principal symbol is the homogeneous polynomial, or matrix-valued polynomial, obtained from the highest-order derivative terms of a differential operator by replacing derivatives with covector variables.

Differential operator: A linear or nonlinear operator built from derivatives of functions or sections. The principal symbol extracts the highest-order part of this broader class of operators.
Elliptic regularity: The principle that solutions of elliptic equations are smoother than their initial formulation may suggest. Ellipticity of the principal symbol underlies the regularity estimates.
Tangent space: The vector space of infinitesimal directions at a point of a smooth manifold. Cotangent vectors used by the symbol are dual to tangent directions.
Subprincipal symbol: The next-order symbol term that supplements a differential or pseudodifferential operator’s principal symbol. It captures lower-order information omitted by the principal symbol.
Principal part: The sum of the highest-order terms in a differential operator. Replacing its derivatives by covector variables produces the principal symbol.
Hyperbolic partial differential equation: A partial differential equation whose characteristic structure supports finite-speed wave propagation. Its characteristic symbol identifies the wavefront directions along which disturbances travel.
Homogeneous polynomial: A polynomial whose monomials all have the same total degree. The scalar principal symbol is homogeneous of degree equal to the operator’s order.
Full symbol: A symbol representation containing the principal and lower-order contributions of an operator. Unlike the principal symbol, it can retain the operator’s full asymptotic data.
Cotangent space: The vector space of linear functionals on a tangent space at a point. The symbol’s covector variables belong to the cotangent space.
Wavefront set: A set recording where a distribution is singular and in which cotangent directions its singularities occur. Principal symbols govern how singularities propagate through solutions.

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