vectorveld

Vectorveld, in Dutch mathematics often rendered as vector field in English, is a mathematical construct that assigns to every point in a space a vector. Formally, if D is a domain in n-dimensional Euclidean space R^n, a vectorveld is a function F: D → R^n. At each point x ∈ D, the vector F(x) describes a quantity with both magnitude and direction. In two dimensions, F(x,y) = (P(x,y), Q(x,y)); in three dimensions, F(x,y,z) = (P(x,y,z), Q(x,y,z), R(x,y,z)). Vectorvelden can also depend on time, written as F: D × T → R^n, representing, for example, a changing velocity field.

Common operations on vectorvelden include divergence, curl, and gradient relations. The divergence div F measures net outflow from a point, while the curl ∇×F (in three dimensions) measures rotation. The gradient ∇φ of a scalar field φ yields a vector field pointing in the direction of greatest increase. The Jacobian matrix JF describes first-order changes of the field, and directional derivatives describe change along a specified direction. A vectorveld is called conservative if F = ∇φ for some scalar potential φ; in such cases line integrals depend only on endpoints, and the work done by the field along a path equals the potential difference.

Visualization and interpretation play a key role in understanding vectorvelden. Field lines or streamlines show the direction of flow, and arrow plots on a grid convey both magnitude and direction.

Applications are wide. Vectorvelden model physical fields such as gravity, electricity, and magnetism; describe fluid velocities in aerodynamics and hydrodynamics; represent wind fields in meteorology; serve in computer graphics as displacement or texture fields; and underpin potential-field methods in robotics and path planning.