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Covariant differentiation on 2-sphere

  1. May 22, 2013 #1
    Consider a 2-sphere S2 with coordinates xμ=(θ,[itex]\phi[/itex]) and metric ds2=dθ2+sin2θ d[itex]\phi[/itex]2 and a vector [itex]\vec{V}[/itex] with components Vμ=(0,1). Calculate the following quantities.

    θ[itex]\phi[/itex]Vθ

    [itex]\phi[/itex]θVθ
     
  2. jcsd
  3. May 22, 2013 #2
    [Mods should move this to calculus and beyond, methinks.]

    I believe the formula for this case would be ##\nabla_{i}\mathbb{e}_j = \left\{\begin{array}\\{ij}\\{k}\end{array}\right\}\mathbb{e}_k##. However, you've shown no work, so I can't help you.
     
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