A vector field associates a vector to every point in space; the vectors may change from point to point. Vector fields are often used in physics, for instance to indicate the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.
In the rigorous mathematical treatment, vector fields are defined on manifolds: a vector field is a section of the manifold's tangent bundle. While the underlying manifold is often the 2-dimensional or 3-dimensional Euclidean space (in which case the tangent space is equal to the same Euclidean space), other manifolds are also useful: describing the wind distribution on the surface of the Earth for instance requires a vector field on the sphere, a 2-dimensional manifold; the spacetime of relativity is a 4-dimensional manifold; and phase spaces of complicated physical systems are often modeled as high dimensional manifolds with a vector field indicating how the system changes over time.
Vector fields should be compared to scalar fields, which associate a number or scalar to every point in space (or every point of some manifold).
The gradient of a scalar field is a vector field. The derivatives of a vector field using a scalar product or a cross product, resulting in a scalar field or another vector field, are called the divergence and curl respectively.
Common vector fields