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As per the title, I'm just looking for a reference with a convenient derivation (or at minimum, description) of the Euclidean metric in hyperspherical coordinates. The specific cases of polar or 3-dimensional spherical coordinates are not helpful.

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I don't know about a reference, but I can show you how to construct it inductively. We have $$g_{Euc} = dr^2 + r^2g_{S^{n-1}},$$ where $g_{S^{n-1}}$ is the round metric on the $n-1$ sphere. So it remains to construct $g_{S^n}$ inductively. Well, $$g_{S^n} = d\rho^2 + \sin^2(\rho)g_{S^{n-1}}.$$

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