1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Isotropic rank 3 pseudotensor help

  1. Mar 2, 2009 #1
    Can anybody show me how any isotropic rank 3 pseudotensor can be written as

    [itex]a_{ijk}=\lambda \epsilon_{ijk}[/itex]

    for the isotropic rank 2 tensor case [i.e. [itex]a_{ij}=\lamda \delta_{ij}[/itex] ], my notes prove it by considering an example i.e. a rotation by [itex]\frac{\pi}{2}[/itex] radians about the z axis.
     
  2. jcsd
  3. Mar 3, 2009 #2
    Re: Tensors

    anyone?
     
  4. Mar 4, 2009 #3
    Re: Tensors

    No help yet? It's not clear to me how you're using indices. Are you writing vectors with lower indices?

    Secondly, are you defining pseudotensors as tensors that change sign under a reflection of the coordinates such as (x,y,z)-->(-x,y,z)? There are two definitions of a pseudotensor.

    Edit: Actually, it should be obvious from your question that you are using the first definition.

    So the claim is that the only pseudotensors, whos elements remain unchanged under a general rotation, are equal to the totally antisymmetric tensor multiplied by some scaling factor.
     
    Last edited: Mar 5, 2009
  5. Mar 5, 2009 #4
    Re: Tensors

    yes.
    i guess then i meant that such a pseudotensor is proprotional to the levi civita rank 3 pseudotensor.

    how does one actually show that though?
     
  6. Mar 5, 2009 #5
    Re: Tensors

    They're not giving you much to go on, are they?

    The easy problem (1) is to show that the elements of [itex]\lambda \epsilon_{ijk}[/itex] are unchanged under othonormal transformations. The hard part is (2) showing that the elements of all rank 3 pseudotensors that remain unchanged are of the form [itex]\lambda \epsilon_{ijk}[/itex].

    The only difference between tensors and pseudovectors in this proof is that you have to eventually account for an optional reflection of coordinates.

    I would tackle the first problem first:


    Tensors transform as the product of vectors.

    An orthornormal transformation in three dimensions can be obtained as the product of the transformations that rotate vectors about the X, Y and Z axis.
     
    Last edited: Mar 5, 2009
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Isotropic rank 3 pseudotensor help
  1. Calculus 3 hw help (Replies: 2)

Loading...