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Jacobian matrix determinant vanishes

  1. Mar 11, 2012 #1
    What exactly does it mean when the determinant of a Jacobian matrix vanishes? Does that imply that the coordinate transformation is not a good one?

    How do you know if you coordinate transformation is a good one or a bad one?
     
  2. jcsd
  3. Mar 11, 2012 #2
    Let me explain further. This is the particular transformation I am looking at:

    [itex]\left(
    \begin{array}{c}
    x(t) \\
    y(t) \\
    z(t)
    \end{array}
    \right)= R_{z}(\gamma) R_{x}(\beta) R_{z}(\alpha)\left(
    \begin{array}{c}
    x_o \\
    y_o \\
    z_o
    \end{array}
    \right)[/itex],

    here, the vector [itex](x_{o},y_{o},z_{o})[/itex] is just the initial positions of small pieces of a solid cube, and the angles [itex]\alpha(t)[/itex],[itex]\beta(t)[/itex],[itex]\gamma(t)[/itex] are the Euler angles (all functions of time).

    When I expand this out, and form the Jacobian matrix, it's determinant vanishes.
     
  4. Mar 11, 2012 #3
    Probably there is a mistake in your calculation. In the standard forms, the determinant of Rx , Ry and Rz all are 1.0 , for any angles. So is the determinant of their product.
     
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