Pregunta: (3) Evaluate the Jacobian ∂(x,y,z)/∂(ρ,ϕ,θ), where x=ρsinϕcosθ,y=ρsinϕsinθ, and z=ρcosϕ

$\mathrm{x}=\rho \sin \varphi \cos \theta$ , $\mathrm{y}=\rho \sin \varphi \sin \theta$ , $\mathrm{z}=\rho \cos \varphi$

$\frac{\partial (\mathrm{x},\mathrm{y},\mathrm{z})}{\partial (\rho ,\varphi ,\theta )}$ = $\begin{array}{c}det\left[\begin{array}{ccc}\frac{\partial x}{\partial \rho }& \frac{\partial x}{\partial \varphi }& \frac{\partial x}{\partial \theta }\\ \frac{\partial y}{\partial \rho }& \frac{\delta y}{\partial \varphi }& \frac{\partial y}{\partial \theta }\\ \frac{\partial z}{\partial \rho }& \frac{\partial z}{\partial \varphi }& \frac{\partial z}{\partial \theta }\end{array}\right]\end{array}$

$\begin{array}{c}=det\left[\begin{array}{ccc}\sin \varphi & \rho \cos \varphi .\cos \theta & -\rho \sin \varphi .\sin \theta \\ \sin \varphi .\sin \theta & \rho \cos \varphi .\sin \theta & \rho \sin \varphi .\cos \theta \\ \cos \varphi & -\rho \sin \varphi & 0\end{array}\right]\end{array}$