# Covariant derivative summation convention help

## 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?

nrqed
Homework Helper
Gold Member

## 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"?

Chestermiller
Mentor
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.

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