Differentialoperatorer

Differentialoperatorer is not a standard term in mathematics; the conventional terminology is differential operator (or differential operators in the plural). A differential operator is a linear operator that maps functions or sections of a vector bundle to other functions or sections by combining derivatives with smoothly varying coefficients. In Euclidean space, these operators are local: the value of the output at a point depends only on the behavior of the input near that point.

Locally, a differential operator of order at most m on functions on R^n can be written in the form

L = sum_{|α| ≤ m} a_α(x) ∂^α,

where α is a multi-index, ∂^α denotes partial derivatives, and the coefficients a_α(x) are smooth. The order m is the highest total degree of derivatives appearing. With vector-valued inputs, the operator acts componentwise via matrix-valued or bundle-valued coefficients. Coefficients may be constant or vary with position.

The principal symbol of a differential operator captures its leading behavior and plays a key role in analysis. For L as above, the principal symbol at (x, ξ) is σ_L(x, ξ) = sum_{|α|=m} a_α(x) ξ^α, a homogeneous polynomial in the cotangent variable ξ. The operator is called elliptic if σ_L(x, ξ) ≠ 0 for all ξ ≠ 0. Classification also includes hyperbolic and parabolic types, according to the signature of the principal symbol, which informs wave propagation and diffusion properties.

Extensions and applications: differential operators extend naturally to distributions and act on sections of vector bundles over manifolds. They form the building blocks of partial differential equations, enable the construction of fundamental solutions and Green’s functions, and connect to broader frameworks such as adjoints, parametrices, and, more generally, pseudodifferential operators.