4D angular coordinate system and corresponding hypervolume element

[areslagae]

I am looking for a 4D angular coordinate system (radius and three angles) and its corresponding "hypervolume element".

2D: polar coordinates - dA = r dr dtheta
3D: spherical coordinates - dV = r^2 sin(phi) dphi dtheta dr
4D: ?

 

[yyat]

4-D spherical coordinates:

x1 = r sin(theta1) sin(theta2) cos(phi)
x2 = r sin(theta1) sin(theta2) sin(phi)
x3 = r sin(theta1) cos(theta2)
x4 = r cos(theta1)

from which you can compute the hypervolume element.

 

[areslagae]

Thanks.

Is "4-D spherical coordinates" established terminology?
Is this the "standard" transformation? I guess there are other ones?

 

[yyat]

Thanks.

Is "4-D spherical coordinates" established terminology?
Is this the "standard" transformation? I guess there are other ones?

I just checked wikipedia. There, the generalization to arbitrary dimension is called http://en.wikipedia.org/wiki/Hypersphere#Hyperspherical_coordinates". They are the same that I gave you in the 4-D case except for the order of the coordinates.
I don't know what the standard term in the case n=4 is.