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I always have trouble when working in spherical coordinates.

Given $|V|$ and the cartesian components $V_x,V_y,V_z$ I want to find the spherical components $V_r,V_\theta,V_\phi$. As far as I understand $V_r=|V|$, but how do I find the other components?

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  • $\begingroup$ What is $|V|$? Is it a generic subset of $\mathbb R^3$ ? $\endgroup$
    – Vajra
    Commented Jul 13, 2021 at 6:57
  • $\begingroup$ @Vajra $|V|=\sqrt{V_x^2+V_y^2+V_z^2}$ $\endgroup$
    – mattiav27
    Commented Jul 13, 2021 at 6:58
  • $\begingroup$ So $|V|$ is a real number but it's not clear to me what $V_x,V_y,V_z$ are. $\endgroup$
    – Vajra
    Commented Jul 13, 2021 at 7:00
  • $\begingroup$ @Vajra They are the components of the vector in cartesian coordinate $\endgroup$
    – mattiav27
    Commented Jul 13, 2021 at 7:03

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Trivially, $\theta=\cos^{-1}{V_z \over \sqrt{V_x^2+V_y^2}}$ and $\phi=\tan^{-1}{V_y \over V_x} (+\pi)$ but these are not `components of a vector' as you can't add two vectors by adding these 'components', or multiply them by a scalar.

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