From: hees@itp.uni-frankfurt.de   
      
   It's only that vectors or covectors and in general any tensor of any   
   rank do not transform at all under coordinate transformations (i.e.,   
   diffeomorphisms). What transforms are the basis vectors and the   
   corresponding dual base and correspondingly the components of the   
   tensors wrt. these bases. I tried to summarize this briefly for vectors   
   and covectors in my posting.   
      
   Note that in physics you usually have more structure in your manifolds.   
   E.g., in GR you assume a pseudo-Riemannian manifold, i.e., a manifold   
   with a fundamental form (of Lorentz signature (1,3) or (3,1) depending   
   on your sign convention) with the torsion-free compatible affine   
   connection. Then you can also canonically (i.e., independent of the use   
   of bases and cobases) identify vectors and covectors, as is usually done   
   by physicists.   
      
   On 09/08/2024 06:15, Mikko wrote:   
   > On 2024-08-07 11:37:02 +0000, the moderator said:   
   >   
   >> I think Stefan is using "tangent vector" and "cotangent vector"   
   >> in the sense of differential geometry and tensor calculus.  In   
   >> this usage, these phrases describe how a vector (a.k.a a rank-1   
   >> tensor) transforms under a change of coordintes: a tangent vector   
   >> (a.k.a a "contravariant vector") is a vector which transforms the   
   >> same way a coordinate position $x^i$ does, while a cotangent vector   
   >> (a.k.a a "covariant vector") is a vector which transforms the same   
   >> way a partial derivative operator $\partial / \partial x^i$ does.   
   >   
   > Thank you. That makes sense.   
   >   
   > --=20   
   > Mikko   
      
   --   
   Hendrik van Hees   
   Goethe University (Institute for Theoretical Physics)   
   D-60438 Frankfurt am Main   
   http://itp.uni-frankfurt.de/~hees/   
      
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