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Covariant derivative of vector fields on the sphere

  • #1
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?
 

Answers and Replies

  • #2
TSny
Homework Helper
Gold Member
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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.
 
  • #3
132
4
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.
 
  • #4
TSny
Homework Helper
Gold Member
12,232
2,745
That looks good.
 
  • #5
132
4
That looks good.
Thanks for the reminder!
 

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