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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
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