Almost everyone outside of mathematics uses the term vector field only in the Euclidean plane or three-space. In this context, a vector field is a function F, defined on some subset U of R2 or R3, whose value F(x) at each point x ∈ U is a vector in R2 or R3 respectively. You imagine each value F(x) as an arrow, rooted at x, whose length and direction are that of F(x). Usually there is a tacit assumption that F is smooth, or at least differentiable once or twice. For example, the electric field E induced by a distribution of charge in space is a vector field.

Electric field is a bad example with respect to smoothness, however, as physics students frequently work with models involving point charges or other nonsmooth charge densities, and the resulting fields are differentiable almost everywhere, but have singularities. The practical resolution is that physicists learn a collection of informal rules for handling functions which aren't everywhere smooth, but behave well enough overall; and mathematicians either close their eyes, or learn distribution theory, which is a sophisticated form of calculus capable of handling most singularities that arise in practice.

In physics it is common to restrict the terms vector and vector field to describing only those quantities which are invariantly defined, that is, which have "real physical meaning". Some things which are formally vector quantities, such as the angular momentum of classical mechanics, don't satisfy this condition. In classical mechanics if we have a point mass with momentum p, and its displacement vector from a given point O in space is r, then we say that the angular momentum of the particle about O is L = r × p, where × is the cross product of vectors in space. If you change coordinates by reflecting in a plane, so that the sign of one of the coordinates is reversed, the cross product also changes sign, so L now points the opposite way even though the physical situation has not changed. To a physicist this means that L is not a vector, but a so-called pseudovector. Similarly you can have fields of pseudovectors, which change sign on an orientation-reversing change of coordinates.

To a mathematician the term vector field denotes a slightly fancier construction. A vector field X on a manifold M is a smooth section of the tangent bundle TM, that is for each point m ∈ M a choice of a tangent vector X(x) ∈ TmM so that, considered as a map of manifolds, X: M → TM is smooth.

I have never seen an exposition of mechanics in these terms, but I believe that the division into vectors and pseudovectors can be expressed in terms of the mathematically natural operations on the vector bundles associated to the base space R3:

For instance, the cross product a × b is really ρ∗(λa ∧ λb) where λ: TM → T*M (for "lower indices") and ρ: T*M → TM (for "raise indices") are the duality isomorphisms, ∧: ΛiT*M ⊗ ΛjT*M → Λi+jT*M is the exterior product, and ∗: ΛjT*M → Λ3-jT*M is the Hodge star.

Most people, even most physicists, don't care about all this machinery; only a few of us pure math nerds are bothered by the fact that "pseudovector" doesn't have a precise mathematical definition. I'm only in the process of working it out for myself.

To learn more about the calculus of vector fields from an applied point of view I recommend a little book entitled div grad curl and all that by Harry M. Schey. For the pure mathematics perspective turn to Calculus on manifolds by Michael Spivak, or Volume III of Jean Dieudonné's Treatise on analysis.

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