Clear[x, y, z, r, theta, phi]
x[r_, theta_, phi_] = r Sin[theta] Cos[phi] ;
y[r_, theta_, phi_] = r Sin[theta] Sin[phi] ;
z[r_, theta_, phi_] = r Cos[theta]  ;

(* check hand calculations *)
(*FullSimplify[Solve[{x == x[r, theta, phi], y ==y[r, theta, phi], z ==z[r, theta, phi] },{r, theta, phi}] , x > 0 && y > 0 && z > 0]*)
r[x_, y_, z_] = Sqrt[x^2 + y^2 + z^2] ;
theta[x_, y_, z_] = ArcCos[z/r[x, y, z]] ;
phi[x_, y_, z_] = ArcTan[y/x] ;

(* check hand calculation *)
pr[x_, y_, z_, px_, py_, pz_] = FullSimplify[
m  D[r[x[t], y[t], z[t]], t] /. x'[t] -> px/m /. y'[t] -> py/m  /.  z'[t] -> pz/m /. x[t] -> x /. y[t] -> y /. z[t] -> z, x > 0 && y > 0 && z > 0]

ptheta[x_, y_, z_, px_, py_, pz_] = FullSimplify[m  r[x, y, z]^2 D[theta[x[t], y[t], z[t]], t] /. x'[t] -> px/m /. y'[t] -> py/m  /.  z'[t] -> pz/m /. x[t] -> x /. y[t] -> y /. z[t] -> z, x > 0 && y > 0 && z > 0]

pphi[x_, y_, z_, px_, py_, pz_] = 
FullSimplify[m  r[x, y, z]^2  Sin[theta[x, y, z]]^2  D[phi[x[t], y[t], z[t]], t] /. x'[t] -> px/m /. y'[t] -> py/m  /.  z'[t] -> pz/m /. x[t] -> x /. y[t] -> y /. z[t] -> z, x > 0 && y > 0 && z > 0]

j[x_, y_, z_, px_, py_, pz_] = 
FullSimplify[D[{r[x, y, z], theta[x, y, z], phi[x, y, z], pr[x, y, z], ptheta[x, y, z], pphi[x, y, z]}, {{x, y, z, px, py, pz}}] , x > 0 &&y > 9 && z > 0 && px > 0 && py > 0 && pz > 0]
j[x, y, z, (Subscript[p, x]), (Subscript[p, y]), (Subscript[p, z])] // TraditionalForm
Det[ j[x, y, z, (Subscript[p, x]), (Subscript[p, y]), (Subscript[p, z])]] // FullSimplify