Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Cartesian Tensors and transformation matrix

  1. Jul 28, 2007 #1
    I was just reading chapter on Cartesian tensors and came across equation for transformation matrix as function of basic vectors. I just do not get it and cannot find a derivation. I am too old to learn Latex, I uploaded a word document with the equation. Thanks, Howard

    Attached Files:

  2. jcsd
  3. Jul 29, 2007 #2

    Lij=cos(e'i,ej)=e'i dot ej.
  4. Jul 29, 2007 #3
    I am looking at page 930-931 Of Riley, Hobson, Bence.

    They go from:

    e'j,= Sij ei

    X'i = (S[tex]^{-1}[/tex])ij Xj

    Define L as inverse of matrix S

    X'i = Lij Xj, since rotations of coordinate axes are rigid, transformation matrix L is orthogonal, thus the inverse transformation is

    Xi = Lji X'j and

    Lik Ljk = [tex]\delta[/tex]ij and Lki Lkj = [tex]\delta[/tex]ij

    "furthermore, in terms of basis vectors of the primed and unprimed Cartesian coordinate system, the transformation matrix is given by

    Lij = e'i dot ej

    I understand the cosine formula for dot product but do not see how the transformation matrix follows from this argument. I am starting to get the Latex, but all the i's and j's are of course subscripts.

    Last edited: Jul 29, 2007
  5. Jul 29, 2007 #4
    using x'ie'i=xjej ; If you start from e'j,= Sij ei, then Sij is some matrix which is defined.x'je'j=xiei or x'jSij ei=xiei ,i.e. Sij x'j=xi, i.e.x'j=[Sinverse]jixi , noting how matrix operation is applicable[again e.g.by dotting by dotting e'k for both sides then you get x'k=xjLkj]...

    Some property Lki Lkj = delij, should be remembered once for all.Then you can get everything else.
    Last edited: Jul 29, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Cartesian Tensors and transformation matrix
  1. Matrix Transformation (Replies: 8)

  2. Matrix Transform (Replies: 8)