Why this doesn't transform properly?

Question

We are in $\mathbb R^n $, with a tensor field of components $T_\nu$, and being $e_\mu$ the vectors of the basis: $e_\mu \equiv \partial_\mu$, then I'm asked to show that $\partial_\mu T_\nu$ can't be, in general, the components of a tensor. I'm looking at how it transforms under a basis change, but I don't see anything wrong.

Answer 1

Since you consider $\partial_\mu T_\nu$ a smooth function, and therefore can be the component of many vectors.

Moreover it will be the component of the $\nabla T$ derivative as long as the frame $\{\partial_\mu\}_{\mu=1}^n$ is geodesic, i.e., $\nabla_{\partial_\nu} \partial_\mu=0$, which is the case in $\mathbb{R}^n$ with the usual coordinates.