Tensor transformations for change of coordinate system

  1. In school I've always learned that tensor transformations took the form of:

    [tex]\mathbf{Q'}=\mathbf{M} \times \mathbf{Q} \times \mathbf{M}^T [/tex]

    However, in all the recent papers I've been reading. They've been doing the transformation as:

    [tex]\mathbf{Q'}= \frac {\mathbf{M} \times \mathbf{Q} \times \mathbf{M}^T}{det(\mathbf{M})}[/tex]

    Where Q is the tensor in question and M is the transformation matrix and M^T is the transpose of M.

    Does anyone know why the difference?
  2. jcsd
  3. Ben Niehoff

    Ben Niehoff 1,766
    Science Advisor
    Gold Member

    Such on object is properly called a "tensor density". Specifically, it is a "tensor density of weight -1", since the determinant appears with the power -1.
  4. I can't seem to find a clear definition of "tensor density" online. How does this differ (or provide an advantage) (practically, not mathematically) from a regular coordinate transformation? FYI, I'm trying to follow the transformation of a anisotropic density (actual matter density) in a paper.
  5. Ben Niehoff

    Ben Niehoff 1,766
    Science Advisor
    Gold Member

    Did you even try Googling it? For your claim that you "can't find it", I have serious doubts.
  6. Of course I tried googling it. I didn't say I didn't find ANY definitions, I said I couldn't find a CLEAR definition which included information pertaining to the entirety of the second sentence of my last post with an emphasis on the parenthetical parts. The online definitions basically say its a coordinate transformation with a weight to it based on a Jacobian determinant. Big whoop, those definitions tell me nothing about why it's used over a normal transformations. Why use a weight of -1 rather than 1000?
    Last edited: Aug 15, 2011
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