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I Example of *Non* Cartesian Vector/Tensor (not the coordinate s

  1. Sep 10, 2015 #1
    ...system, I mean as in the Cartesian Vector/Tensor definition.

    I get that if you have two mutually orthogonal basises which are theta degrees apart and the transformation from one basis to the other follows the same as a rotation by theta degrees i.e:

    V'i = Rij Vj

    then it is a Cartesian vector.

    Does that mean if you wanted to change the basis from a mutually orthogonal one to a non mutually orthogonal one V would no longer be a Cartesian vector? Are there any other (better) examples of non Cartesian Vectors?

    Thanks
     
  2. jcsd
  3. Sep 10, 2015 #2

    Orodruin

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    Objects which transform as general vectors (or tensors) are always Cartesian vectors (tensors) as the Cartesian coordinate transformations are a special case of the more general coordinate transformations. The reverse is not true, you can have a Cartesian vector which is not a vector under general coordinate transformations as it can transform as a vector under Cartesian transformations but this does not imply that it will transform as a vector under more general coordinate transformations.

    I would not call any object a "non-Cartesian vector" as this seems to imply that it is not a vector under Cartesian coordinate transformations. The more general concept would just be "vectors".
     
  4. Sep 10, 2015 #3
    Does this just mean all proper/polar vectors are always Cartesian vectors?

    and
    are proper rotations (determinate +1) ?

    You can have a Cartesian vector which transforms as a vector under proper rotations but does not transform as a vector under improper rotations (determinant -1). I.e you can have a pseudo which is also a Cartesian vector? How can it then be that the transformation law for a Cartesian vector is V'i = Rij Vj ? Is a pseudo vector only a Cartesian vector under proper (Cartesian) transformation?
     
    Last edited: Sep 10, 2015
  5. Sep 10, 2015 #4
    Does anyone have any examples? It would really help
     
  6. Sep 10, 2015 #5

    mfb

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    What exactly do you mean by "Cartesian Vector"? The representation of an arbitrary vector in a cartesian coordinate system?
     
  7. Sep 11, 2015 #6
  8. Sep 11, 2015 #7

    Orodruin

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    Yes, but all Cartesian tensors are not tensors under general transformations. An example is the Levi-Civita symbol ##\varepsilon_{ijk}##. It is a Cartesian (pseudo) tensor, but not a tensor under general transformations. (It is, however, a tensor density.)

    Under proper rotations, pseudo vectors and vectors behave in the same way. This is not restricted to Cartesian vectors.
     
  9. Apr 10, 2016 #8
    Is there any proper definition for non cartesian coordinates?
     
  10. Apr 11, 2016 #9

    Orodruin

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    Why would there not be? Any coordinate system which is not Cartesian is non-Cartesian.
     
  11. Apr 11, 2016 #10
    if we talk about non cartesian tensor then what does its mean?
     
  12. Apr 11, 2016 #11

    Orodruin

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    You really should then just be talking about "tensor". At what level have you learned what a Cartesian tensor is? (Your answer here will make it easier to gaugke the level of the response.)
     
  13. Apr 12, 2016 #12
    cartesian coordinates are those coordinates which indicate the location of point relative to its fixed refrence point.
    In case of three points, they all intersect at right triangle at the origin.
     
  14. Apr 13, 2016 #13

    Orodruin

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    This is just the coordinates. The question was about tensors.
     
  15. Apr 13, 2016 #14
    Kindly tell me the right answer about non cartesian tensor because tomorrow will last date for my assignment submition.
     
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