I'm finding it difficult to find any non-geometrical derivation of coordinate conversions from cartisan to spherical.
I can understand the derivations geometrically, because I can visualize the construction. However, if I would want to find n-spherical coordinates conversions, how would I do so?
In other words, how do I go from 2D:
\begin{align} x & =r\cos\theta \\[6pt] y & =r\sin\theta \\[6pt] \end{align}
to 3D:
\begin{align} z & =r\cos\theta \\[6pt] y & =r\sin\theta\sin\phi \\[6pt] x & =r\sin\theta\cos\phi \\[6pt] \end{align}
to 4D:
\begin{align} w & =r\cos\theta \\[6pt] z & =r\sin\theta\cos\phi \\[6pt] y & =r\sin\theta\sin\phi\sin\psi \\[6pt] x & =r\sin\theta\sin\phi\cos\psi \\[6pt] \end{align}
and etc..
Ok, I can see a pattern to emerge and can guess that 5D is:
\begin{align} v & =r\cos\theta \\[6pt] w & =r\sin\theta\cos\phi \\[6pt] z & =r\sin\theta\sin\phi\cos\psi \\[6pt] y & =r\sin\theta\sin\phi\sin\psi\sin\delta \\[6pt] x & =r\sin\theta\sin\phi\sin\psi\cos\delta \\[6pt] \end{align}
But why?