I'm pretty new to tensors in differential geometry and I have a basic question about the notation used. In general a vector field $X$ can be expressed as
$$X=\sum_{i=1}^n X^i \partial_i,$$
where $X^i = v(x^i)$ is the vector-coordinate and $\partial_i = \frac{\partial}{\partial x^i}|_p, \ v \in T_p (M)$ is the basis.
Likewise, a one form $\theta$ can be written as
$\theta = \sum_{i=1}^n \theta_i dx^i$,
again, where $\theta_i = \theta(\partial_i)$ is the coordinate and $dx^i$ is the (dual) basis (please correct me if any of the above is wrong). What confuses me is that an (r,s)-tensor $A$ can then be written as
$A=\sum A_{j_1,...,j_s}^{i_1,...,i_r} \partial_{i_1} \otimes ... \otimes \partial_{i_r} \otimes dx^{j_1} \otimes ... \otimes dx^{j_s} $,
Where $j_1,...,j_s$ are my covariant components (vector fields) and $i_1,...,i_r$ are my contravariant components (one-forms). Why does the notation seemingly change with respect to the basis? Here, it looks like $\partial_{i_1} \otimes...\otimes \partial_{i_r}$ is my contravariant (dual) basis and $dx^{j_1} \otimes ... \otimes dx^{j_s}$ is my covariant basis? Is this just a notation tradition or am I missing something? Can anyone explain exactly what the basis terms do in terms of operations in the last expression, in case I am off? Thanks a million.