Vector field

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In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space ${\displaystyle \mathbb {R} ^{n}}$.[1] A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.
A vector field is a special case of a vector-valued function, whose domain's dimension has no relation to the dimension of its range; for example, the position vector of a space curve is defined only for smaller subset of the ambient space. Likewise, n coordinates, a vector field on a domain in n-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law (covariance and contravariance of vectors) in passing from one coordinate system to the other.