vektorfelt

A vector field, or vektorfelt, is a mathematical object that assigns a vector to every point in a region of space. Formally, a vector field F on a domain D subset of R^n is a function F: D -> R^n. In two dimensions, F(x, y) = (P(x, y), Q(x, y)). The value F(x, y) is the vector representing, for example, a velocity, force, or magnetic field at the point (x, y).

Vector fields can be visualized by arrows at each point, with the arrow's direction indicating the vector and length indicating magnitude. They are used to model many physical and geometric situations, such as fluid flow, electric or gravitational fields, and flow lines or streamlines.

For differentiable vector fields, several local operators are defined. The Jacobian matrix, consisting of partial derivatives, describes how the field changes in space. The divergence div F measures the net outflow of the field from a point; curl F measures rotation in three dimensions. For a scalar field φ, the gradient ∇φ is a vector field pointing in the direction of greatest increase of φ, and a gradient field is conservative.

Conservative vector fields have path-independent line integrals and can be written as the gradient of a potential function. Divergence-free fields have zero divergence and model incompressible flows. Key theorems such as Green's, Stokes', and the divergence theorem relate line and surface integrals of vector fields to these differential operators. Vector fields are central in physics, engineering, and computer graphics.