Discussion
Match the coordinate system with its element of volume. 1. Cartesian 2. Cylindrical 3. Spherical P. \( r\,dr\,d\theta\,dz \) Q. \( dx\,dy\,dz \) R. \( r^{2}\sin\phi\,dr\,d\phi\,d\theta \)
Answer: B
Cartesian coordinates need no factor, all three edges being lengths already. Cylindrical carries one factor of \( r \) for the single angle. Spherical carries \( r^{2}\sin\phi \), one power of the radius for each of its two angles and a sine because circles of latitude shrink towards the poles
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