View source for Vector field

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A term which is usually understood to mean a function of points in some space $X$ whose values are vectors (cf. [[Vector|Vector]]), defined for this space in some way.

In the classical [[Vector calculus|vector calculus]] it is a subset of a Euclidean space that plays the part of $X$, while the vector field represents directed segments applied at the points of this subset. For instance, the collection of unit-length vectors tangent or normal to a smooth curve (surface) is a vector field on it.

If $X$ is an abstractly specified differentiable manifold, a vector field is understood to mean a tangent vector field, i.e. a function that associates to each point of $X$ an (invariantly constructed) vector tangent to $X$. If $X$ is finite-dimensional, the vector field is equivalently defined as a collection of univalent, contravariant tensors, which are depending on the points.

In the general case a vector field is interpreted as a function defined on $X$ with values in a [[Vector space|vector space]] $P$ associated with $X$ in some way; it differs from an arbitrary vector function in that $P$ is defined with respect to $X$  "internally"  rather than as a  "superstructure"  over $X$. A section of a [[Vector bundle|vector bundle]] with base $X$ is also considered to be a vector field.



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Cf. also [[Vector field on a manifold|Vector field on a manifold]].