For a vector bundle (E,π,M) let ϕ:M↦E be a section of π, x∈M and u=ϕ(x). The vertical differential of the section ϕ at point u∈E is the map: dvuϕ:TuE↦Vuπ
In coordinates on
E (xi,uα) we write;
dvuϕ=(duα−∂ϕα∂xidxi)⊗∂∂uα
Apparently it is
obvious from this that
dvuϕ depends only on the first order jet space
j1xϕ.
What is Vuπ in this case? It is clearly related to the jet manifold J1π whose total space is the product T∗M⊗EVπ . But I don't really understand what an associated vector bundle is!
References:
- C.M. Campos, Geometric Methods in Classical Field Theory and Continuous Media, pages 24-25.
This post imported from StackExchange Mathematics at 2015-08-12 17:47 (UTC), posted by SE-user Janet the Physicist