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Covariant derivative summation convention help

  • #1

Homework Statement


Assume that you want to the derivative of a vector V with respect to a component Zk, the derivative is then ∂ViZi/∂Zk=Zi∂Vi/∂Zk+ViZi/∂Zk = Zi∂Vi/∂Zk+ViΓmikZm Now why is it that I can change m to i and i to j in ViΓmikZm?
 

Answers and Replies

  • #2
nrqed
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Homework Statement


Assume that you want to the derivative of a vector V with respect to a component Zk, the derivative is then ∂ViZi/∂Zk=Zi∂Vi/∂Zk+ViZi/∂Zk = Zi∂Vi/∂Zk+ViΓmikZm Now why is it that I can change m to i and i to j in ViΓmikZm?
What do you mean by "I can change m to i and i to j"?
 
  • #3
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What are the boldface ##Z_m##s. Are they supposed to be coordinate basis vectors? If so, and you are really expressing the covariant derivative in this way, then there should be ##\mathbf{Z_k}## also, so that there are dyadic products of coordinate basis vectors. The gradient of a vector is a 2nd order tensor.
 
  • #5
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1. Now why is it that I can change m to i and i to j in ViΓmikZm?
Because in that expression "m" is a dummy index and so is "i". The name of a dummy index can be changed without changing the meaning of the equation
 

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