Vector field

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In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.

Vector fields are often used in physics to model, for example, 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, (tangent) vector fields are defined on manifolds as sections of a manifold's tangent bundle. They are one kind of tensor field on the manifold.

Contents

Definition

Vector fields on subsets of Euclidean space

Given a subset S in Rn, a vector field is represented by a vector-valued function  V: S \to \mathbf{R}^n in standard Cartesian coordinates (x1, ..., xn). If S is an open set, then V is a continuous function provided that each component of V is continuous, and more generally V is Ck vector field if each component V is k times continuously differentiable.

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