The discussion focuses on covariant differentiation on the 2-sphere \( S^2 \) with coordinates \( x^\mu = (\theta, \phi) \) and the metric \( ds^2 = d\theta^2 + \sin^2 \theta \, d\phi^2 \). Participants are tasked with calculating the covariant derivatives \( \nabla_\theta \nabla_\phi V^\theta \) and \( \nabla_\phi \nabla_\theta V^\theta \) for the vector \( \vec{V} \) with components \( V^\mu = (0, 1) \). The discussion highlights the importance of showing work in calculations, as one participant emphasizes the need for detailed steps to provide assistance.
PREREQUISITESMathematicians, physicists, and students engaged in advanced calculus, differential geometry, or general relativity who seek to deepen their understanding of covariant differentiation on curved surfaces.
[Mods should move this to calculus and beyond, methinks.]felixphysics said: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