A term which is usually understood to mean a function of points in some space $X$ whose values are vectors (cf. Vector), defined for this space in some way.
In the classical 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 $P$ 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 with base $X$ is also considered to be a vector field.
Cf. also Vector field on a manifold.
Vector field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_field&oldid=31705