J

Any section σ(x) = (xµ, yi(x)) induces a section of J kB defined by

jkσ (x) = (xµ, yi(x), ∂µyi(x), . . . , ∂µ1...µk yi(x))

which is called the k-jet prolongation of σ . Any bundle morphism ) : B → B ′ (pro-

jecting over a diffeomorphism φ : M → M ′) induces a bundle morphism J k) : J kB → J kB ′ defined by

jk)(jkx σ ) = jkf (x)() ◦ σ ◦ f −1)

which is called the k-jet prolongation of ). Analogously, any projectable vector field X

over B induces a projectable vector field jkX over J kB which is called the k-jet prolongation of X.