I am trying to do a four-dimensional change of variables problem, and I am working with hyperspherical coordinates. For $(x,y,z,w)=\varphi(\rho, \varphi, \theta, \lambda)$, I have
$x= \rho \sin\varphi \sin\theta \sin\lambda$
$y=\rho \sin\varphi \sin\theta \cos\lambda$
$z=\rho \sin\varphi \cos\theta$
$w=\rho \cos\varphi$
I am not completely sure if this conversion is correct, so feedback on that would be appreciated. Now, I need to find the determinant of the Jacobian of the polar coordinates. Here is the matrix I have
\begin{vmatrix} \sin\varphi \sin\theta \sin\lambda & \rho \cos\varphi \sin\theta \sin\lambda & \rho \sin\varphi \cos\theta \sin\lambda & \rho \sin\varphi \sin\theta \cos\lambda\\ \sin\varphi \sin\theta \cos\lambda & \rho \cos\varphi \sin\theta \cos\lambda & \rho \sin\varphi \cos\theta \cos\lambda & -\rho \sin\varphi \sin\theta \sin\lambda\\ \sin\varphi \cos\theta & \rho \cos\varphi \cos\theta & -\rho \sin\varphi \sin\theta & 0\\ \cos\varphi & -\rho \sin\varphi & 0 & 0 \end{vmatrix}
I am not going to write all of my calculations out, but my final answer is $-\rho^3 \sin^2\varphi \sin\theta$. Does that seem right, or am I doing something wrong?