Scale factors and the Jacobian Matrix

[Andromon]

So here is my problem: I'm very bad when using coordinates others than cartesian ones, and I know taking Mechanics of Lagrange and Hamilton and I fin difficult to find the velocities expressions in curvilinear coordinates.

So here is my question: is there anyway to relate the scale factors (and everything else, from radius vector expression to operators expressoins) just using the Jacobian or derivates matrix? Like you can do in Thermodynamics

I have not seen it done in any book, but I cannot see why not. Also, if it can't be done, what will be a quickest method or passing from everything in one coordinate frame to another?

Thank you.

 

[Mark44]

So here is my question: is there anyway to relate the scale factors (and everything else, from radius vector expression to operators expressoins) just using the Jacobian or derivates matrix?

Yes. The scale factor is the determinant of the Jacobian matrix, the matrix with the first partial derivatives of the components in one coordinate system with respect to the components in the other coordinate system.