Thread: question on pushforward on tangent vectors

in my book tangent vectors are defined as equivalence classes of curves that are tangent to the given vector. let σ be a curve on a manifold and h*([σ]) is defined to be [h o σ]. v1 = [σ1] and v2 = [σ2]. show that the pushforward is linear: h*(v1 + v2) = h*(v1) + h*(v2) and h*(rv1) = rh*(v1).

i am starting this problem with showing the first part h*(v1 + v2) = h*(v1) + h*(v2) and i am having trouble. earlier my book defines v1 + v2 = [φ^-1 (φ o σ1 + φ o σ2)] where φ is a coordinate function.

i have that h*(v1+v2) = [h(φ^-1 (φ o σ1 + φ o σ2))] but i don't know how to get from there to h*(v1) + h*(v2) = [h o σ1] + [h o σ2]. if φ^-1 was linear then it would work but that doesn't seem to be the case all the time however.

Hey oblixps. I'll gladly help you with your question if you'll be so kind so as to reword it in LaTeX so that I can read it without getting a major headache.

Thanks!