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I have read that contra-variant and co-variat vectors have different transformation properties , which distinguish them, yet at the same time I have read that a vector can have contra-variant and co-variant components. If a vector can have contra-variant and co-variant components, then isn't this vector both contra-variant and co-variant? But isn't this impossible?

Bob Bobby
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  • Co- and contravariant vectors are related by the musical isomorphism. – ACuriousMind Jan 11 '15 at 22:14
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    contravariant and covariant vector is IMHO extremely bad terminology. A vector is just a vector, an arrow on a manifold pointing somewhere. We like to describe these objects by using numbers, hence we express vectors in terms of components. We can do this either in their original vector spaces, or by taking them to the dual space, whenever we have an object that allows us to do it in a more or less "standard" way (like a metric). – Phoenix87 Jan 11 '15 at 22:15
  • related: http://physics.stackexchange.com/q/105347/58382, http://physics.stackexchange.com/q/91593/58382, http://physics.stackexchange.com/q/37820/58382 – glS Jan 11 '15 at 22:17
  • @Phoenix87 I don't know, I think that description ignores the reason physical vectors are different from mathematical vectors, namely transformations. – David Z Jan 11 '15 at 22:21
  • @DavidZ: sorry, I don't get your comment – Phoenix87 Jan 11 '15 at 22:40
  • Bob, higher rank tensors have both co- and contra-variant components, e.g., $R^{\alpha}_{\beta \mu \nu}$, but I don't see how this is possible with a rank 1 tensor. – Alfred Centauri Jan 11 '15 at 22:54

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