$\nabla_\mu X^{\nu} = \delta^{\nu}_{\mu}$ ? Covariant derivative of coordinate

Question

Let the coordinate on a $d$-dimensional manifold be given by $X^{\mu}$. I wish to know what is $\nabla_{\nu}X^{\mu} = ?$ i.e. the covariant derivative of a coordinate ? I understand that this maybe an ill-defined object, so please just humor me.

Answer 1

A coordinate is a scalar function of the point, so it makes sense to take its derivative, a gradient in this case. If you move along the corresponding coordinate line, the rate of increase is clearly $1$. If you move along any other coordinate line, it does not change and the derivative is zero. That is why the result is $\delta_\mu^\nu$.