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The differential length vector in spherical coordinates is

  • A.\( r \, dr \, \mathbf{a}_r + d\theta \, \mathbf{a}_\theta + \sin\theta \, d\phi \, \mathbf{a}_\phi \)
  • B.\( dr \, \mathbf{a}_r + r \, d\theta \, \mathbf{a}_\theta + r \sin\theta \, d\phi \, \mathbf{a}_\phi \)
  • C.\( dr \, \mathbf{a}_r + d\theta \, \mathbf{a}_\theta + d\phi \, \mathbf{a}_\phi \)
  • D.\( dr \, \mathbf{a}_r + r \, d\theta \, \mathbf{a}_\theta + r \, d\phi \, \mathbf{a}_\phi \)

Answer: B

Each component must be a length, so each angular differential is multiplied by the radius of the circle it sweeps. For the azimuthal angle that radius is \( r \sin\theta \), the distance from the z axis rather than from the origin.

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