So, I'm trying to understand the proof of Cartan's test given in "Exterior Differential Systems", by R. L. Bryant et al.. There's 3 statements I'm not able to justify:
$\textbf{Theorem 1.11}$ (Cartan's Test): Let $\mathcal{I} \subset \Omega^{+}(M)$ be an ideal wich contais no non-zero forms of degree $0$. Let $(0)_z \subset E_1 \subset ... \subset E_n \subset T_zM$ be an integral flag of $\mathcal{I}$ and, for each $k < n$, let $c_k$ be the codimension of $H(E_k)$ in $T_zM$. Then $\mathcal{V_n}(\mathcal{I}) \subset G_n(TM)$ is of codimension at least $c_0 + ... + c_{n-1}$ at $E_n$. Moreover, each $E_k$ is regular for all $ k < n$ if and only if $E_n$ has a neighborhood $U$ in $G_n(TM)$ so that $\mathcal{V}_n(\mathcal{I}) \cap U$ is a smooth manifold of codimension $c_0 + ... + c_{n-1}$ in $U$.
1) Why is there a polar sequence $\varphi^1,...,\varphi^{c_{n-1}}$ such that $du^{a}(v) = \varphi(v,\frac{\partial}{\partial x^1},..., \frac{\partial}{\partial x^{\lambda(a)}})$ for all $v \in T_z M$?
2) Assuming 1, it follows that $(\varphi^a)_z = (du^a \wedge dx^1 \wedge ... \wedge dx^{\lambda(a)})_z + (\psi^a)_z$, where $\psi^a$ is as described in the proof, but how can I ensure that it holds for points nearby $z$?
3) Assuming 2, why $F^{a}_j = p^a_j + P^a_j + Q^a_j$ as described in the proof? I was guessing that the terms described in $\textbf{(i)}$ would be part of $P_j^a$, but i can't see why it only depends on the variables $\{ p_i^a \mid (i,a) \ \text{is principal} \ \}$ (and $(x^i, y^j)$, of course)
Thanks in advance.