The covariant derivative of a vector $A^{\mu}$ at a point $x$ is defined as

$$D_z A^{\mu}=\partial_zA^{\mu}+\Gamma^{\mu}_{\rho\sigma}(x)\partial_{z}x^{\rho}A^{\sigma}$$

where Greek symbols are spacetime (dimension $d$) and $z$ is the index of a worldsheet coordinate $\sigma$.

Why doesn't the vector $A^{\mu}$ act as a scalar with respect to the $z$ derivative?
How can I prove this?

Assume $A^{\mu}$ depends on $x^\mu(\sigma^b)$, i.e. $A^{\mu}(x^\mu(\sigma^b))$.

This post imported from StackExchange Physics at 2014-11-26 10:50 (UTC), posted by SE-user sol0invictus