The page reads:
Define the ring $\Omega(M)$ to be the direct sum of the rings of $l$-forms on $M$, for all $l$. If $\Delta$ is a $k$-dimensional distribution on $M$, then $\mathscr{I}(\Delta)\subset\Omega(M)$ will denote the subring generated by the set of all forms $\omega$ with the property that (if $\omega$ has degree $l$) $$ \omega\left( X_1,\ldots,X_l \right) =0\quad \text{whenever }X_1,\ldots,X_l\text{ belong to }\Delta .$$ It is clear that $\omega_1+\omega_2\in\mathscr{I}(\Delta)$ if $\omega_1,\omega_2\in\mathscr{I}(\Delta)$, and that $\epsilon\wedge\omega\in\mathscr{I}(\Delta)$ if $\omega\in\mathscr{I}(\Delta)$ [thus, $\mathscr{I}(\Delta)$ is an ideal in the ring $\Omega(M)$]. Locally, the ideal $\mathscr{I}(\Delta)$ is generated by $n-k$ independent $1$-forms $\omega ^{k+1},\ldots,\omega ^{n}$. In fact, around any point $p\in M$ we can choose a coordinate system $(x,U)$ so that $$ \left.\frac{\partial }{\partial x^{1}} \right|_p,\ldots,\left.\frac{\partial }{\partial x^{k}} \right|_p\ \ \text{span }\Delta _p .$$ Then $$ dx^{1}(p)\wedge\cdots\wedge dx^{k}(p)\ \ \text{is non-zero on }\Delta _p .$$ By continuity, the same is true for $q$ sufficiently close to $p$, which by Corollary 4 implies that $dx^{1}(q),\ldots,dx^{k}(q)$ are linearly independent in $\Delta _q$. Therefore, there are $C^{\infty}$ functions $f_{\beta}^{\alpha}$ such that $$ dx^{\alpha}(q)=\sum_{\beta=1}^{k} f_{\beta}^{\alpha}(q)dx^{\beta}(q)\ \ \textit{restricted to }\Delta _q\quad \alpha =k+1,\ldots,n .$$ We can therefore let $$ \omega ^{\alpha}=dx^{\alpha}-\sum_{\beta=1}^{k} f_{\beta}^{\alpha}dx^{\beta} .$$
My question is why are $f_\alpha^\beta$ not $0$? Since $\{dx^1, ..., dx^n\}$ is a dual basis to $\{\frac{\partial}{\partial x^1}, ..., \frac{\partial}{\partial x^n}\}$, wouldn't we simply get $dx^\alpha(\frac{\partial}{\partial x^\beta})=f_\alpha^\beta=0,\; 1\leq \beta\leq k$? In other words, why can't we just take $\omega^\alpha=dx^\alpha,\; (k+1)\leq\alpha\leq n$ as the generators of the ideal in question?
Also, what is this strange letter Spivak uses to denote this ideal, and does this ideal have a name (maybe annihilator of the distribution or something)?