Express in matrix notation the transformation equations for a (i) covariant vector and (ii) a contravariant tensor of rank two, assuming \(N = 3\).

(i) Covariant Vector:

Let \(A_i\) be a covariant vector. The transformation equation is:

\(A'_i = \frac{\partial x^j}{\partial x'^i} A_j\)

In matrix notation (assuming N=3):

\( \begin{bmatrix} A'_1 \\ A'_2 \\ A'_3 \end{bmatrix} = \begin{bmatrix} \frac{\partial x^1}{\partial x'^1} & \frac{\partial x^2}{\partial x'^1} & \frac{\partial x^3}{\partial x'^1} \\ \frac{\partial x^1}{\partial x'^2} & \frac{\partial x^2}{\partial x'^2} & \frac{\partial x^3}{\partial x'^2} \\ \frac{\partial x^1}{\partial x'^3} & \frac{\partial x^2}{\partial x'^3} & \frac{\partial x^3}{\partial x'^3} \end{bmatrix} \begin{bmatrix} A_1 \\ A_2 \\ A_3 \end{bmatrix} \)

(ii) Contravariant Tensor of Rank Two:

Let \(A^{ij}\) be a contravariant tensor of rank two. The transformation equation is:

\(A'^{ij} = \frac{\partial x'^i}{\partial x^k} \frac{\partial x'^j}{\partial x^l} A^{kl}\)

This doesn't easily translate to a concise single matrix equation because of the double indices. A more detailed representation would involve higher-dimensional arrays or tensor notation.