Covariant differentiation on 2-sphere

[felixphysics]

Consider a 2-sphere S² with coordinates x^μ=(θ,$\phi$) and metric ds²=dθ²+sin²θ d$\phi$² and a vector $\vec{V}$ with components V^μ=(0,1). Calculate the following quantities.

∇_θ∇_$\phi$V^θ

∇_$\phi$∇_θV^θ

 

[Mandelbroth]

Consider a 2-sphere S² with coordinates x^μ=(θ,$\phi$) and metric ds²=dθ²+sin²θ d$\phi$² and a vector $\vec{V}$ with components V^μ=(0,1). Calculate the following quantities.

∇_θ∇_$\phi$V^θ

∇_$\phi$∇_θV^θ

[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.