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In cylindrical coordinates the element of volume is

  • A.\( dr\,d\theta\,dz \)
  • B.\( r^{2}\sin\phi\,dr\,d\phi\,d\theta \)
  • C.\( r^{2}\,dr\,d\theta\,dz \)
  • D.\( r\,dr\,d\theta\,dz \)

Answer: D

Cylindrical coordinates are plane polars with \( z \) carried along unchanged, so the plane element \( r\,dr\,d\theta \) is simply multiplied by the height:

\[ dV \;=\; r\,dr\,d\theta\,dz. \]

The form with \( r^{2}\sin\phi \) is the spherical element, where both angles are true angles and two factors of length are needed to convert them

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