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This link (at the end of the page) is deriving the formula to calculate an infinitesimally small volume $dV$ on the surface of a sphere with radius $v$ in velocity space. It states that $dV = dv_xdv_zdv_z$. I have drawn this small volume piece in velocity space myself and deduced the following enter image description here

So I have deduced that $dV = \sin(\theta) \cdot v \cdot d\phi \cdot v \cdot d\theta \cdot dv$. This is also what the link states. However, I really can't see how this is the same as $dV = dv_xdv_zdv_z$. According to my drawing, I have deduced that $$dv_z = \sin(\theta)\cdot v\cdot d\phi\cdot \sin(\phi)$$ $$dv_x = \sin(\theta)\cdot v\cdot d\phi\cdot \cos(\phi)$$ $$dv_y = \sin(\theta) \cdot v \cdot d\theta$$ But I can't equate them to make $dV = \sin(\theta) \cdot v \cdot d\phi \cdot v \cdot d\theta \cdot dv$.

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  • $\begingroup$ you should obtain the metric $G$ for a sphere. if you have the metric your volume element is: $dV=\sqrt{|\det{G}|}\,dv\,d\Phi\,d\theta$ $\endgroup$ – Eli Aug 2 at 16:16
  • $\begingroup$ @Eli Sorry I don't know what you mean. Isn't it possible to deduce it using my derived formulas? $\endgroup$ – JohnnyGui Aug 2 at 16:22
  • $\begingroup$ this is how you can calculate the $dV$ mathematically, but if you don't understand it you have to use your geometrical construct,sorry i can't help you with this $\endgroup$ – Eli Aug 2 at 16:26

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