What is the analogue of spherical coordinates in $n$-dimensions? What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by
$$x_1=r \cos  \theta$$
$$x_2=r \sin  \theta$$
For $n=3$, the analogue would be the ordinary spherical coordinates $r,\theta ,\varphi$, related to the Cartesian coordinates $x_1,x_2,x_3$ by
$$x_1=r \sin  \theta  \cos  \varphi$$
$$x_2=r \sin  \theta  \sin  \varphi$$
$$x_3=r \cos  \theta$$
So these are my questions: Is there an analogue, or several, to spherical coordinates in $n$-dimensions for $n>3$? If there are such analogues, what are they and how are they related to the Cartesian coordinates? Thanks.

Just look at n-sphere. A lecture note from Stony Brook is also available.
You can find it in Fock's paper (Fock, V. (1935).)or in some recent papers- like (Howard, S. "Fundamental Solution of Laplaces Equation
in Hyperspherical Geometry.") or (Jing-Jing, F., Ling, H., & Shi-Jie, Y. (2011). Solutions of laplace equation in n-dimensional spaces. Communications in Theoretical Physics, 56(4), 623.) for advanced studies. 
