Jacobian matrix determinant vanishes

[Demon117]

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?

 

[Demon117]

Let me explain further. This is the particular transformation I am looking at:

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

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

When I expand this out, and form the Jacobian matrix, it's determinant vanishes.

 

[Hassan2]

Probably there is a mistake in your calculation. In the standard forms, the determinant of R_x , R_y and R_z all are 1.0 , for any angles. So is the determinant of their product.