# Vector field

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Vector field given by vectors of the form (-y, x)

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a 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 the manifold's tangent bundle.

## Definition

Given a subset S in Rn a vector field is represented by a vector-valued function $\displaystyle V: S \to \mathbf{R}^n$ in standard Euclidean coordinates (x1, ..., xn). If there is another coordinate system y, then $\displaystyle V_y := \frac{\partial x}{\partial y} V$ is the expression for the same vector field in the new coordinates. In particular a vector field is not a bunch of scalar fields.

We say V is a Ck vector field if V is k times continuously differentiable. A point p in S is called stationary if the vector at that point is zero ($\displaystyle V(p) = 0$ ).

A vector field can be visualized as a n-dimensional space with a n-dimensional vector attached to each point. Given two Ck-vector fields V, W defined on S and a real valued Ck-function f defined on S, the two operations scalar multiplication and vector addition

$\displaystyle (fV)(p) := f(p)V(p)$
$\displaystyle (V+W)(p) := V(p) + W(p)$

define the module of Ck-vector fields over the ring of Ck-functions.