Chapter 14.7, Problem 48ES

(a) Consider the transformation $x=r\cos \theta ,y=r\sin \theta ,z=z$ from cylindrical to rectangular coordinates where $r\ge 0.$ Show that $\frac{\partial \left(x,y,z\right)}{\partial \left(r,\theta ,z\right)}=r$

(b) Consider the transformation $x=\rho \sin \varphi \cos \theta ,y=\rho \sin \varphi \sin \theta ,z=\rho \cos \varphi$ from spherical to rectangular coordinates, where $0\le \varphi \le \pi .$ Show that $\frac{\partial \left(x,y,z\right)}{\partial \left(\rho ,\varphi ,\theta \right)}={\rho }^2\sin \varphi$