Spherical unit vectors in terms of Cartesian unit vectors

Here’s a way to obtain the spherical unit vectors \hat r, \hat \theta and \hat \phi in terms of the Cartesian unit vectors \hat x, \hat y and \hat z. Note that \theta correspond to the zenithal angle while \phi is the azimuthal angle.

Starting from \begin{cases}x=r\cos\phi\sin\theta \\y=r\sin\phi\sin\theta\\z=r\cos\theta\end{cases}    and \vec r = x \hat x + y \hat y + z\hat z, the spherical unit vectors are given by :

\hat \phi = \frac{1}{h_\phi} \frac{\partial \vec r}{\partial \phi} where h_\phi=|\frac{\partial \vec r}{\partial \phi}|=r\sin \theta is the scale factor. So we have that \hat \phi = \frac{1}{r \sin \theta}(-r\sin \phi \sin \theta \hat x + r \sin \theta \cos \phi \hat y) = - \sin \phi \hat x + \cos \phi \hat y.

While \hat \theta = \frac{1}{r}(r \cos\phi\cos\theta \hat x + r \sin\phi \cos\theta \hat y -r\sin\theta \hat z)=\cos\phi\cos\theta \hat x + \sin\phi \cos\theta \hat y -\sin\theta \hat z;

Finally, \hat r=\frac{\partial \vec r}{\partial r}=\cos\phi\sin\theta \hat x + \sin\phi\sin\theta\hat y +\cos\theta \hat z.


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