Vektorelle

Vektorelle is a term used in discussions of vector fields to denote a localized vector component obtained by weighting a reference vector field with a scalar function. In this sense, a vektorelle represents a vector field that is confined to a region of interest, with its magnitude modulated by the weight function.

Formally, let M be a smooth manifold and V0 a smooth vector field on M, and let w: M -> R be a smooth function with compact support. The vektorelle determined by V0 and w is the pointwise product V = w V0. The support of V is contained in the support of w, and its magnitude is controlled by max|w| times the magnitude of V0. If w and V0 are smooth, so is V.

Vektorelles are used in mathematical modeling and numerical simulations to represent localized influences such as forces, fluxes, or sources. They support linear superposition by combining multiple weighted reference fields to create composite localized fields.

Example: in R^2, take V0 ≡ (1,0) and choose a smooth, compactly supported weight w that peaks near the origin. The resulting V = w(x,y)(1,0) is a localized horizontal flow centered at the origin.

Although the term is not widely standardized, vektorelle serves as a convenient descriptive device for localized vector fields and is related to the ideas of weighted vector fields and compactly supported perturbations.