Vector Field

In vector calculus and physics, a vector field is an assignment of a vector to every point in a space, most commonly Euclidean space, and can be visualized on a plane as a collection of arrows, each with its own magnitude and direction, attached to points across the plane. Vector fields are used to model things such as the speed and direction of a moving fluid like wind, or the strength and direction of a force such as gravity or magnetism as it changes from point to point, and the tools of differential and integral calculus extend naturally to them: when a vector field represents force, the line integral along a path gives the work done by that force, and notions such as divergence and curl describe the rate of change of volume and the rotation of a flow respectively. A vector field is a special case of a vector-valued function, and more generally vector fields are defined on differentiable manifolds, spaces that locally resemble Euclidean space but can have more complicated structure overall, where a vector field assigns a tangent vector to each point of the manifold.