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Coordinate transformation

  1. Jan 14, 2015 #1
    V′μ=((∂yμ)/(∂xν))*Vν

    This is a contravariant vector transformation. (Guys I am really sorry for making the formula above looks so incomprehensible as I still new to this.)

    For the y in the partial derivative, is y a function in terms of x? In that sense, is it formula that maps x to y? Is it something like how much units of x correspond to each unit of y?
     
  2. jcsd
  3. Jan 14, 2015 #2

    Nugatory

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    Staff: Mentor

    Yes, each ##y^i## is a function of the ##x^i##, and vice versa.

    A common and easy example is Cartesian and polar coordinates applied to the two-dimensional Euclidean plane. Using your notation, and with the ##y^i## as the Cartesian coordinates, the ##x^i## as the polar coordinates, and ##i## takes on the values 0 and 1 because this is a two-dimensional space:

    ##y^0=x^0cos(x^1)##; you're more used to seeing ##x=rcos\theta##
    ##y^1=x^0sin(x^1)##; you're more used to seeing ##y=rsin\theta##

    If you have the components of a vector in polar coordinates ##V^\mu##, the formula you posted will give you the components of the same vector in Cartesian coordinates ##V'^\mu## (using the Einstein summation convention, of course).
     
  4. Jan 14, 2015 #3
    Thanks! This really help a lot.
     
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