Coordinate transformations

Coordinate transformation is given by the set of functions $x^\prime=x^\prime(x)$. It is often convenient to add prime to the index, i.e write $x^{\alpha^\prime}\left(x^\alpha\right)$. This way $alpha^\prime$ not only means that the vector is considered in other frame, but also $alpha^\prime$ being different index than $\alpha$ we save on Greek letters. Otherwise we would have to write ${x^\prime}^\beta \left(x^\alpha\right)$ to designate the dependence of $\beta$ primed coordinate on $\alpha$ original one. This is especially useful for differentiation, so we can write ${A^\alpha}_{,\beta^\prime}$ to say the $\alpha$'s component of vector A in original frame is differentiated wrt to $\beta$ coordinate in another, primed, system.

We consider coordinate transformation invertible, i.e., there exist the inverse functions $x^\alpha \left(x^{\alpha^\prime}\right)$. In this case, using chain differentiation rule

$$\frac{\partial x^\alpha}{\partial x^{\alpha^\prime}} \frac{... ...{\partial x^\alpha}{\partial x^\beta} = {\delta^\alpha}_\beta$$ (4)

the latter step following from condition that coordinates in the same system are independent on each other. Similarly,

$$\frac{\partial x^{\alpha^\prime}}{\partial x^\alpha} \frac{... ...l x^{\beta^\prime}} = {\delta^{\alpha^\prime}}_{\beta^\prime}$$ (5)

$\frac{\partial x^\alpha}{\partial x^{\alpha^\prime}}$ can be viewed as square matrices, but they are not tensors, since they are not objects defined in one particular coordinate system (actually they mix two coordinate systems). Coordinates $x^\alpha$'s themselves do not constitute a vector, and when they are viewed as functions of another coordinate set, the transformation matrix is build from usual derivatives of these functions wrt their arguments. Quite similar with what you have to deal with when, for example, you change variables in a multidimensional integral.