Contra-Variant Vector Transform: Taking Partials

[exmarine]

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?

 

[HallsofIvy]

Those are NOT "vectors" or "tensors"- they are tensor or vector valued functions. That is they are functions that assign a tensor or vector to every point in space-time. Just as you do not take derivtives of numbers, but of functions, so the derivative is a function that can be evaluated at any point in space-time.