Covariant derivative of vector fields on the sphere

[Whitehole]

Homework Statement

Given two vector fields $W_ρ$ and $U^ρ$ on the sphere (with ρ = θ, φ), calculate $D_v W_ρ$ and $D_v U^ρ$. As a small check, show that $(D_v W_ρ)U^ρ + W_ρ(D_v U^ρ) = ∂_v(W_ρU^ρ)$

Homework Equations

$D_vW_ρ = ∂_vW_ρ - \Gamma_{vρ}^σ W_σ$
$D_vU^ρ = ∂_vU^ρ + \Gamma_{vσ}^ρ U^σ$

$\Gamma_{φφ}^θ = -sinθ~cosθ, ~~~~~\Gamma_{θφ}^φ = \frac{cosθ}{sinθ}$

The Attempt at a Solution

I've calculated all the problem asked as

$D_θW_θ = ∂_θW_θ, ~~~~~~~ D_θW_φ = ∂_θW_φ - \frac{cosθ}{sinθ}W_φ$

$D_φW_θ = ∂_φW_θ - \frac{cosθ}{sinθ}W_φ, ~~~~~~~ D_φW_φ = ∂_φW_φ + sinθ~cosθ~W_θ$$D_θU^θ = ∂_θU^θ, ~~~~~~~ D_θU^φ = ∂_θU^φ + \frac{cosθ}{sinθ}U^φ$

$D_φU^θ = ∂_φU^θ - sinθ~cosθ~U^φ, ~~~~~~~ D_φU^φ = ∂_φU^φ + \frac{cosθ}{sinθ}U^θ$

For the check,

$(D_θW_φ)U^φ + W_φ(D_θU^φ) = (∂_θW_φ - \frac{cosθ}{sinθ}W_φ)U^φ + W_φ(∂_θU^φ + \frac{cosθ}{sinθ}U^φ)$

The second and the fourth term obviously cancels so it satisfies the equality.

I'm stuck here,

$(D_φW_θ)U^θ + W_θ(D_φU^θ) = (∂_φW_θ - \frac{cosθ}{sinθ}W_φ)U^θ + W_θ(∂_φU^θ - sinθ~cosθ~U^φ)$

The second and the fourth term do not cancel each other, also U and W doesn't match each others subscript/superscript. What I'm thinking is to transform either U or W so that the subscript/superscript will match but I'm not sure how to do it. Any suggestions?

 

[TSny]

For the check,

$(D_θW_φ)U^φ + W_φ(D_θU^φ) = (∂_θW_φ - \frac{cosθ}{sinθ}W_φ)U^φ + W_φ(∂_θU^φ + \frac{cosθ}{sinθ}U^φ)$

The left hand side is not the complete expression that you want to look at. You need to write out the left side of $(D_v W_ρ)U^ρ + W_ρ(D_v U^ρ) = ∂_v(W_ρU^ρ)$. Note the summation over the index $\rho$ while the index $v$ is held fixed.

 

[Whitehole]

The left hand side is not the complete expression that you want to look at. You need to write out the left side of $(D_v W_ρ)U^ρ + W_ρ(D_v U^ρ) = ∂_v(W_ρU^ρ)$. Note the summation over the index $\rho$ while the index $v$ is held fixed.

What do you mean it is not complete? Do you mean, the right hand side? I forgot that it is a summation on ρ.

$(D_θW_θ)U^θ + (D_θW_φ)U^φ + W_θ(D_θU^θ) + W_φ(D_θU^φ) = ∂_θW_θU^θ + ∂_θW_φU^φ - \frac{cosθ}{sinθ}W_φU^φ + W_θ∂_θU^θ + W_φ∂_θU^φ + \frac{cosθ}{sinθ}W_φU^φ$

It should be like this right? Then for the case where $D_φ$, I can already see that the terms that I thought will not cancel, should already cancel.

 

[TSny]

That looks good.

 

[Whitehole]

That looks good.

Thanks for the reminder!