Vectorfield

A vector field on a domain D in Euclidean space is a function that assigns to every point x in D a vector F(x) in the same space, typically indicating direction and magnitude at that point. In R^n, F maps D to R^n. It can be thought of as an arrow attached to each point.

In R^3, common examples include velocity fields, which give the velocity of a moving fluid at each location, and force fields, which assign a force vector to positions in space. Electric and magnetic fields in physics are vector fields as well, though in electromagnetism more general field objects exist.

Vector fields are studied with concepts from calculus. If F is differentiable, one can consider its Jacobian, which describes how F changes; the divergence measures local outward flux, and the curl measures local rotation in three dimensions. A line integral of F along a curve computes work done by F on a particle moving along that curve. If F is a gradient of a scalar function, F = ∇φ, the field is conservative and line integrals depend only on endpoints.

An integral curve of F is a solution to the differential equation x'(t) = F(x(t)), representing the path of a particle moving under the field. The associated flow maps an initial point to its position at time t.

Vector fields have wide applications in physics, engineering, fluid dynamics, meteorology, and computer graphics. In graph theory or discrete settings, vector fields can be defined on graphs by assigning a vector to each vertex or edge.