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In this Wikipedia page the third covariant basis vector in the spherical coordinate system (when using the physics $r, \theta, \varphi$ convention) is:

$$ e_{\varphi} = R \cos \theta (-\sin \varphi \quad \cos \varphi \quad 0)$$

Is this wrong? In both this answer and Pavel Grinfeld's textbook (section 9.2.8) it appears that it should be $\sin \theta$ instead of $\cos \theta$.

blz
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    Depends how $\theta$ is defined. If, as usual, it's angle from the north pole, then it's obviously wrong since when $\theta = \frac{\pi}{2}$ we have $e_\varphi = 0$. – Charles Hudgins Sep 16 '23 at 18:02
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    Wikipedia says $\theta$ is the latitude. As we know the north pole as at latitude $90^\circ$ while in physics convention it is $\theta=0,.$ Nothing is wrong. Just a different $\theta,.$ – Kurt G. Sep 16 '23 at 18:25

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