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## Main Question or Discussion Point

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

dx[itex]^{\mu}[/itex]=x[itex]^{\mu}_{,\nu}[/itex]dx[itex]^{\nu}[/itex]

A[itex]^{\mu}[/itex]=x[itex]^{\mu}_{,\nu}[/itex]A[itex]^{\nu}[/itex]

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?

dx[itex]^{\mu}[/itex]=x[itex]^{\mu}_{,\nu}[/itex]dx[itex]^{\nu}[/itex]

A[itex]^{\mu}[/itex]=x[itex]^{\mu}_{,\nu}[/itex]A[itex]^{\nu}[/itex]

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?