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.