Another contra-variant vector question

  • Thread starter exmarine
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  • #1
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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?
 

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  • #2
HallsofIvy
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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.
 

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