Generalized vector fields over a * bundle* are not vector fields on the bundle in the standard sense; nevertheless, one can drag sections along them and thence define their Lie derivative. The formal Lie derivative on a bundle may be seen as a generalized vector field. Furthermore, generalized vector fields are objects suitable to describe generalized symmetries.

Let B = (B, M, π; F) be a bundle, with local fibered coordinates (x^{μ}; y^{i}). Let us consider the pull-back of the tangent bundle τ_{B}: T_{B} → B along the map π^{k}_{0}: J^{k}B → B:

A generalized vector field of order k over B is a section Ξ of the fibre bundle π^{* }: π^{k}_{0 }: ^{*}T_{B} → J^{k}B, i.e.

for each section σ: M → B, one can define Ξ_{σ} = i ○ Ξ ○ j^{k}σ: M → TB, which is a vector field over the section σ. Generalized vector fields of order k = 0 are ordinary vector fields over B. Locally, Ξ(x^{μ}, y^{i}, …, y^{i}_{μ1,…μk}) is given the form:

Ξ = ξ^{μ}(x^{μ}, y^{i}, …, y^{i}_{μ1,…μk})∂_{μ} + ξ^{i}(x^{μ}, y^{i}, …, y^{i}_{μ1,…μk})∂_{i}

which, for k ≠ 0, is not an ordinary vector field on B due to the dependence of the components (ξ^{μ}, ξ^{i}) on the derivative of fields. Once one computes it on a section σ, then the pulled-back components depend just on the basic coordinates (x^{μ}) so that Ξ_{σ} is a vector field over the section σ, in the standard sense. Thus, generalized vector fields over B do not preserve the fiber structure of B.

A generalized projectable vector field of order k over the bundle B is a generalized vector field Ξ over B which projects on to an ordinary vector field ξ = ξ^{μ}(x)∂_{μ} on the base. Locally, a generalized projectable vector field over B is in the form:

Ξ = ξ^{μ}(x^{μ})∂_{μ} + ξ^{i}(x^{μ}, y^{i}, …, y^{i}_{μ1,…μk})∂_{i}

As a particular case, one can define generalized vertical vector fields (of order k) over B, which are locally of the form:

Ξ = ξ^{i}(x^{μ}, y^{i}, …, y^{i}_{μ1,…μk})∂_{i}

In particular, for any section σ of B and any generalized vertical vector field Ξ over B, one can define a vertical vector field over σ given by:

Ξ_{σ} = ξ^{i}(x^{μ}, σ^{i}(x),…, ∂_{μ1,…, μk}σ^{i}(x))∂_{i}

If Ξ = ξ^{μ}∂_{μ} + ξ^{i}∂_{i} is a generalized projectable vector field, then Ξ_{(v)} = (ξ^{i} – y^{i}_{μ}ξ^{μ})∂_{i} = ξ^{i}_{(v)}∂_{i} is a generalized vertical vector field, where Ξ_{(v)} is called the vertical part of Ξ.

If σ’: ℜ x M → B is a smooth map such that for any fixed s ∈ ℜ σ_{s}(x) = σ'(s, x): M → B is a global section of B. The map σ’ as well as the family {σ_{s}}, is then called a 1-parameter family of sections. In other words, a suitable restriction of the family σ_{s}, is a homotopic deformation with s ∈ ℜ of the central section σ = σ_{0}. Often one restricts it to a finite (open) interval, conventionally (- 1, 1) (or (-ε, ε) if “small” deformations are considered). Analogous definitions are given for the homotopic families of sections over a fixed open subset U ⊆ M or on some domain D ⊂ M (possibly with values fixed at the boundary ∂D, together with any number of their derivatives).

A 1-parameter family of sections σ_{s} is Lie-dragged along a generalized projectable vector field Ξ iff

(Ξ_{(v)})_{σs} = d/ds σ_{s}

thus dragging the section.