I am getting confused with naming things covariant or contravariant. I am reading Barret O Neil's book on semi riemannian geometry with applications to relativity, and I need some help with the indices.
On page 35 it says that a tensor field
\begin{equation} A:\mathcal{X}^*(M)^r \times \mathcal{X}(M)^s\rightarrow\mathcal{F}(M) \end{equation}
A is a multilinear machine which when fed $r$ one-forms $\theta^1,...,\theta^r$ and $s$ vector fields $X_1,...,X_s$, produces a real-valued function
\begin{equation} f=A(\theta^1,...,\theta^r,X_1,...,X_s)\in\mathcal{F}(M), \end{equation}
so I see that a one-form is some object that has an upper index.
However, on page 37 it says that a tensor of type $(0,s)$ is said to be covariant (lower index), such as real-valued functions and one-forms, while a tensor of type $(r,0)$ is said to be contravariant (upper index) like vector fields. So this tells that a one-form should have a lower index.
Also, on page 38 it says that $\alpha^1,...,\alpha^r\in T_pM^*$ are one-forms because they live in the dual space, while the vector fields are $x_1,...,x_s\in T_pM$, but on page 39 again the one-form is $\theta=\sum\theta_k dx^k$ (with lower index) and the vector field $X=\sum X^i\partial_i$ (with upper index).
I guess it has to do with whether you refer to the component or the element of the basis, but I would like some confirmation because this seems rather non-trivial but no clarification is made throughout the book, so I might be missing something. If that is the case, then why isn't it more useful to write directly the elements of the basis $A(dx^1,...,dx^r,\partial_1,...,\partial_{s}) $ instead of $A(\theta^1,...,\theta^r,X_1,...,X_s)$, which is a notation usually employed to refer to the components?
The confusion lies in the context. In $$f=A(\theta^1,\theta^2,...,\theta^r,X_1,...,X_s)\in\mathcal{F}(M),$$ the meaning of the "$1$" in $\theta^1$ is "the first one-form of a list of one-forms", the meaning of the "$2$" in $\theta^2$ is "the second one-form of a list of one-forms" and so on.
While in $$\theta=\sum\theta_k dx^k$$ the meaning is "the component $\theta_k$ is a covariant quantity" and "the basis element $dx^k$ is a contravariant quantity".