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Covariant and Contravariant Tensors

  1. Dec 13, 2014 #1
    Not sure where to post this thread.

    That being said, can someone explain to me simply what covariant and contravariant tensors are and how covariant and contravariant transformation works? My understanding of it from googling these two mathematical concepts is that when you change the basis of these two tensors, the scale of the resultant varies differently between the two. Please use simple to understand examples.

    Thank you.
    Last edited: Dec 13, 2014
  2. jcsd
  3. Dec 13, 2014 #2


    Staff: Mentor

    this may help:


    In general curvilinear coordinates you have two choices for a basis one is the tangent vectors at a given point and the other is the normal vectors. Whats confusing is that sometimes they line up and there is no distinction visually when they are an orthogonal basis.
    Last edited: Dec 13, 2014
  4. Dec 14, 2014 #3

    Stephen Tashi

    User Avatar
    Science Advisor

    We could begin with Cartesian tensors. In that context, do you understand how a change of coordinates applied to a column vector can be represented by multiplying the vector on the left by an invertible square matrix ?
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