I didn't really understand what it means for coordinates to be covariant and contravariant. I looked at wikipedia but this confused me even more. So I tried to do some calculation :
Let $V$ be a vector space and let $(e_i)_i$ and $(f_i)_i$ be 2 bases of $V$. Also consider the dual basis which are respectively $(\epsilon_i)_i$ and $(\phi_i)_i$. Now if the coordinate transform between $(e_i)_i$ and $(f_i)_i$ is $f_i = \sum_i P_{ik}e_k$, I found out that in the dual bases, the coordinate transform is $\phi^i=\sum_k (P^{-T})_{ik} \epsilon^k$.
Also given these transformations, I have for $x=\sum_i x^i e_i=\sum_i \widehat{x}_i f_i$ that $\widehat{x^i} = \sum_{k} (P^{-T})_{ik}x^k$ (i.e. transformation varies against the rule of the transformation of $(e_i)_i \rightarrow (f_i)_i$, i.e. contravariant transformation rule for those types of coordinates) and for $g=\sum_i g_i \epsilon^i = \sum_i \widehat{g_i} \phi^i$ that $\widehat{g_i}=\sum_k P_{ik} g^k$. (transformation varies with the rule of the transformation as $(e_i)_i \rightarrow (f_i)_i$, i.e. covariant transformation rule for those type of coordiantes).
Now for a more complicated example, le $T : V\times V \times V^*\to \mathbb{R}$ be a tensor that is 2-contravariant and 1-covariant. And expressed it in the bases : $T=\sum_{i,j,k}T^{i,j}_k e_i \otimes e_j \otimes \epsilon^k = \sum_{i,j,k}\widehat{T^{i,j}_k} f_i \otimes f_j \otimes \phi^k$ then the upper (contravariant) indices will be transformed with $P^{-T}$ and the lower indices (covariant) will be transformed with $P$ right? I.e. we get $\widehat{T^{i,j}_k}=\sum_q P_{kq} \sum_j (P^{-T})_{jm} \sum_i (P^{-T})_{il}T^{l,m}_q$.
Do I understand this concept properly?
N.B : Also, $e_i$ is with lower index (covariant), because its transformation is $f_i=\sum_k P_{ik}e_k$ which varies with the basis transformation (because it actually is the basis transformation!) and $\epsilon^i$ is with upper index (contravariant) because the transformation of these vectors $\phi^i=\sum_k(P^{-T})_{ik}\epsilon^k$ varies with $(P^{-T})$, i.e. against the basis transformation.