4D angular coordinate system and corresponding hypervolume element

AI Thread Summary
A 4D angular coordinate system is being sought, specifically one that includes a radius and three angles, along with its hypervolume element. The discussion references established 2D and 3D coordinate systems, noting that the proposed 4D spherical coordinates can be derived from these. The coordinates provided allow for the computation of the hypervolume element, although there is uncertainty about the standard terminology for 4D spherical coordinates. The conversation also mentions a Wikipedia reference that discusses hyperspherical coordinates in arbitrary dimensions, which aligns with the proposed 4D system. Clarification on the standard term for 4D coordinates remains unresolved.
areslagae
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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: ?
 
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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.
 
Thanks.

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