Covariant derivative summation convention help

In summary, the conversation discusses the derivative of a vector V with respect to a component Zk. The equation for this derivative is ∂ViZi/∂Zk=Zi∂Vi/∂Zk+Vi∂Zi/∂Zk = Zi∂Vi/∂Zk+ViΓmikZm. The question is raised as to why the indices m and i can be interchanged in the term ViΓmikZm. The answer is that in this expression, m and i are dummy indices and can be changed without altering the equation.
  • #1
Mathematicsresear
66
0

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?
 
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  • #2
Mathematicsresear said:

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
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
Mathematicsresear said:
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
 

1. What is the covariant derivative summation convention?

The covariant derivative summation convention is a mathematical notation used in differential geometry and general relativity. It allows for the compact representation of equations involving the covariant derivative, which is a mathematical tool used to differentiate tensors on a curved space.

2. How does the covariant derivative summation convention work?

The covariant derivative summation convention works by representing the covariant derivative of a tensor as the sum of its components multiplied by the metric tensor. This allows for a simpler and more concise representation of equations involving the covariant derivative.

3. What is the advantage of using the covariant derivative summation convention?

The advantage of using the covariant derivative summation convention is that it simplifies and streamlines calculations involving the covariant derivative. It also allows for a more intuitive understanding of the geometric meaning behind these calculations.

4. How is the covariant derivative summation convention different from the standard notation?

The covariant derivative summation convention differs from the standard notation in that it uses the metric tensor to represent the components of the covariant derivative, rather than explicitly writing out each component. This results in a more compact and efficient notation.

5. Are there any limitations to using the covariant derivative summation convention?

While the covariant derivative summation convention is a useful tool, it may not be suitable for all situations. In some cases, it may be more appropriate to use the standard notation to avoid confusion or to accurately represent certain mathematical concepts. Additionally, the covariant derivative summation convention may not be applicable in spaces with non-metric connections.

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