vektorët

Vektorát is a term used in some mathematical literatures to denote a vector-valued field. Broadly, a vektorát assigns to every point in a domain a vector in a corresponding vector space, typically R^n. In standard usage, the domain is an open subset of R^n or a differentiable manifold, and the codomain is R^n or the tangent space at each point.

Formally, if U is a subset of R^n, a vektorát F on U is a function F: U -> R^n. The components F_i are real-valued functions on U. The regularity of F is described by classes such as continuous, C^k, or smooth, depending on the differentiability of its components.

Common examples include velocity fields in fluid dynamics, where F(x) gives the velocity of the fluid at x, and gravitational or electric fields, where F(x) expresses force per unit mass or per unit charge at x. A vektorát can also be viewed as a section of the trivial bundle U x R^n over U.

Analytical tools for a vektorát include the divergence ∇·F, the curl ∇×F in three dimensions, and line and surface integrals of F. These quantities enable the study of circulation, flux, and flow patterns. If F is conservative, it equals the gradient of a scalar potential.

Applications span physics, engineering, computer graphics, and dynamical systems. The concept is closely related to, and sometimes used interchangeably with, the standard notion of a vector field. See also vector field, tensor field, differential geometry, and flow.