Is the Jacobian Equal to the Quotient of Scale Factors?

[Jhenrique]

In somewhere in wikipedia, I found a "shortcut" for compute the jacobian, the formula is: $$\frac{\partial(q_1 , q_2 , q_3)}{\partial (x, y, z)} = h_1 h_2 h_3$$ where q represents the coordinate of other system and h its factor of scale.

I know that this relationship is true. What I'd like of know is if this equation below is true: $$\frac{\partial(q_1 , q_2 , q_3)}{\partial (Q_1, Q_2, Q_3)} = \frac{h_1 h_2 h_3}{H_1 H_2 H_3}$$ where Q represents the coordinate of another system and H its factor of scale.

Is correct to affirm that the jacobian is equal to the quotient between the scale factors?

 

[coquelicot]

What you did is to invent a definition of $$\frac{\partial(q_1 , q_2 , q_3)}{\partial (Q_1, Q_2, Q_3)} .$$ It is not true nor false: it is a definition.