Sorry for this naive question, but I try in this moment to revise my classis. Could you help me please to demonstrate :
$$\frac{\partial T^{\prime\mu}}{\partial x^{\prime \alpha}}=\frac{\partial}{\partial x^{\prime \alpha}}\left(\frac{\partial x^{\prime \mu}}{\partial x^{v}} T^{v}\right)$$
i.e by using a changing of coordinates between $x^{\mu}$ and $x^{\prime\mu}$. $T^{\mu}$ represents the contravariant components of a "vector" (I mean a tensor of rank 1).
That demostration would be done by proving that :
$$T'^{\mu}\text{d}x^{\nu}=T^{\nu}\text{d}x^{\prime\mu}$$
How to justify this equality if I apply a 1-form $\text{d}x^{\mu}$ or another 1-form $\text{d}x^{\prime\mu}$ on a vector $T$ ?