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Contravariant vs Covariant components - misprint?

  1. Jan 16, 2012 #1
    In their article [Integrals in the theory of electron correlations, Annalen der Physik 7, 71] L.Onsager at el. write:

    By resolving the vector [itex]\vec{s}[/itex] into its contravariant components in the oblique coordinate system formed by the vectors [itex]\vec{q}[/itex] and [itex]\vec{Q}[/itex] it is possible to reduce the region of integration (3.11) to a rectangle. The contravariant components of [itex]\vec{s}[/itex] are defined by:

    [itex]s_1 = \frac{\vec{s}\vec{q}}{|q|}, \quad s_2 = \frac{\vec{s}\vec{Q}}{|Q|}[/itex]

    Shouldn't they write "covariant components"?

    PDF of the article can be found here: http://zs.thulb.uni-jena.de/receive/jportal_jparticle_00133463

    Thank you.
     
  2. jcsd
  3. Jan 19, 2012 #2

    Fredrik

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    Why should it be "covariant"? I don't see a reason to mention either of those words. ##s_1## and ##s_2## are just the components of ##\vec s## in the ordered basis ##\Big\langle\frac{\vec q}{|\vec q|},\frac{\vec Q}{|\vec Q|}\Big\rangle##.

    However, if I had to choose one of those words, I'd go with "contravariant", because when ##\{e_i\}## is a basis for the tangent space of a manifold, a tangent vector v can be expressed as ##v=v^i e_i##, and the ##v^i## are said to define a contravariant vector. (I strongly dislike this terminology, and don't understand why people are still using it). The "covariant components of ##\vec v##" in this (horrible) terminology would be components of the 1-form ##g(\vec v,\cdot)## in the dual basis for the cotangent space that's the dual basis of ##\{e_i\}##.

    Also note that if the manifold is Euclidean, the contravariant and covariant components are the same.
     
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