I found inverse transformation from spherical coordinates to cartesian coordinates (on $x>0$, $y>0$ and $z>0$). I have $$ r = w_1(x,y,z) = \sqrt{x^2+y^2+z^2} $$ $$ \theta = w_2(x,y,z) = \arccos\biggl(\frac{z}{\sqrt{x^2+y^2+z^2}}\biggr) $$ $$ \phi = w_3(x,y,z) = \arcsin\biggl(\frac{y}{\sqrt{x^2+y^2}}\biggr) $$
Now I have to find the Jacobian matrix of this application. What is the better way? By direct derivation, I have a problem with $\arccos$ and $\arcsin$ functions. However, inverting the Jacobian matrix of the inverse transformation (cartesian to spherical) is a long task...
Any help?