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
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
- A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (magnitude) of the arrow will be an indication of the wind speed. A "high" on the usual barometric pressure map would then act as a source (arrows pointing away), and a "low" would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas.
- Velocity field of a moving fluid. In this case, a velocity vector is associated to each point in the fluid. In wind tunnels, the fieldlines can be revealed using smoke.
- Magnetic fields. The fieldlines can be revealed using small iron filings.
- Maxwell's equations allow us to use a given set of initial conditions to deduce, for every point in Euclidean space, a magnitude and direction for the force experienced by a charged test particle at that point; the resulting vector field is the electromagnetic field.