- #1

Whitehole

- 132

- 4

## 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?