I am studying the article On the Lagrange-Dirichlet converse in dimension three.
Lemma 2.15. With respect to the coordinate chart corresponding to $θ_1 \neq 0$ and defining $ξ_1 = 1$ for notational convenience, in the region $y_1 \neq 0$ we have:
(1) $(y_j ∂_{{y}_{i}})^∗ = ξ_j ∂_{ξ_i}, \forall i \neq 1.$
(2) $(y_j ∂_{y_1})^∗ = ξ_j ( y_1 ∂_{y_1} − \sum_{i=2}^l ξ_i ∂_{ξ_i}).$
(3) $(∂_{ξ_i})_∗ = y_1 ∂_{y_i}, \forall i \neq 1.$
(4) $(y_1 ∂_{y_1})_∗ = \sum_{i=2}^l y_i ∂_{y_i}.$
My doubts are:
In item (1), what is happening is the pullback of a vector field? How to prove item (1)? I had understood that the pullback should involve a function, for example the pullback of a function f by the vector field V (from what I understood, it would be a precomposition). Now I'm in doubt if there is a need for another function.
In item (3) is the pushout/pushforward of only one of the components of the vector field occurring? How is this possible?
In the article, the author mentions that the proof of this lemma is simple, but I could not prove it, if you have any reference that helps me to understand and prove this lemma, I appreciate it.