Calculate Tensor and its transformations, transformation of derivatives of dependent variables

I had been learning tensor notation for a while and here's what's I have read:

1 Tensor had rank, denote two types covarient or contravariant.

2 $T(\__\alpha,\__\beta,\_\gamma)$ in place naming notion was $=T^{ijk}e_ie_je_k(\__\alpha,\__\beta,\_\gamma)$.

3 Quote from modern classical physics: When doing tensor calculus, coefficient $T^{ijk}$ never changes, what changed was the bases i.e. $T=T^{ijk}e_ie_je_k$ and $T_{ijk}=T(e_i,e_j,e_k)$ so $T^{ijk}$ abbreviates for $T$ in terms of $e_i,e_j,e_k$; and transformation $T_{\alpha\beta\gamma}=T^{ijk}g_{i\alpha}g_{j\beta}g_{k\gamma}$.

4 Each tensor could be written as combination of asymmetry and symmetry parts.

5 $\displaystyle A^i_j=\frac{\partial x^i}{\partial x^l}\frac{\partial x^m}{\partial x^j}A^l_m$ so $\displaystyle g_l^i=\frac{\partial x^i}{\partial x^l}$ and $\displaystyle g^m_j=\frac{\partial x^m}{\partial x^j}$.

Question 1: However, I did have the following question about the exact transformation of the metric.

Suppose I have $V^{ij}=\begin{bmatrix} x_{i1j1} & x_{i1j2} \\ x_{i2j1} & x_{i2j2} \end{bmatrix}= \begin{bmatrix} \sin(t) & \cos(t) \\ \cos(w) & \sin(w*t) \end{bmatrix}$ where $e_i=\{i,j\}$(cartesian coordinates x,y) and $e_j=\{r,\theta\}$(polar coordinates)

In example, how to calculate the transformation $g^\alpha_i$ and $g^\beta_k$ of $V^{\alpha\beta}$ where $e_\beta=\{i,j\}$(cartesian coordinates x,y) and $e_\alpha=\{r,\theta\}$(polar coordinates)?

Question 2, Suppose I had a variable(i.e. time or position) dependent unites, i.e. $e_i=\{(\exp(t)+2,0),(0,\exp(w)+2)\}$(vector pair was written in Cartesian coordinates) and $e_j=\{(t,0),(w,0) \}$ how to calculate the exact expression of $g^i_j$ and $g^j_i$ and $g^{ij}$ and $g_{ij}$? and what were they?

Answer

In general, it was not guaranteed that the transformation could be calculated out.

I.e. by redefine $h=\arcsin(t)$ and expand $\arcsin(t)$ to entire real axis by module. we could find $d t/d\sin(t)$. However it was not for sure that $df(t)/dg(t,x)$ could always obtain a solution. Therefore the bases transformation of arbitrary functions in general could only be solve case by case.