# Another contra-variant vector question

The contra-variant transform seems to be defined by the differential transform from calculus.

dx$^{\mu}$=x$^{\mu}_{,\nu}$dx$^{\nu}$

A$^{\mu}$=x$^{\mu}_{,\nu}$A$^{\nu}$

I am puzzled by this, as the vector / tensor usually has finite components. They span a considerable region of space. So where are the partials to be taken, i.e., at what point in space or space-time?