(n,m) to (n+1,m) derivatives in Wald

In summary, the notation (n,m) to (n+1,m) in Wald refers to the directional derivative of a function along the (n+1)th dimension with all other dimensions held constant at m. It is calculated by taking the limit of the difference quotient as the change in the (n+1)th dimension approaches zero, and is significant in Wald's theory of general relativity as well as in economic and financial models. These derivatives can be negative, indicating a decrease in the function along the (n+1)th dimension, and are used in various practical applications such as physics, economics, and engineering for analyzing the impact of changes in specific variables on overall outcomes and solving optimization and control problems.
  • #1
nickj1
2
0
Could someone please explain the partial and covariant derivatives with upstairs index that appears all of a sudden in Wald in section 4.2 with no explanation? I haven't seen it anywhere else.

Homework Statement



(n,m)→(n+1,m) derivatives

Homework Equations



section 4.2--when the stress tensor is introduced, with downstairs indexes

The Attempt at a Solution



it was mentioned earlier by Wald; just didn't see it
 
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  • #2
Can you perhaps scan the relevant extract ? I don't have Wald handy right now.
 

FAQ: (n,m) to (n+1,m) derivatives in Wald

What does the notation (n,m) to (n+1,m) mean in the context of derivatives in Wald?

The notation (n,m) to (n+1,m) refers to the directional derivative of a function along the (n+1)th dimension, while keeping all other dimensions constant at m. This is also known as the partial derivative.

How is the (n,m) to (n+1,m) derivative calculated?

The (n,m) to (n+1,m) derivative is calculated by taking the limit of the difference quotient as the change in the (n+1)th dimension approaches zero. This can also be expressed in terms of the partial derivative of the function with respect to the (n+1)th dimension.

What is the significance of (n,m) to (n+1,m) derivatives in Wald?

These derivatives are important in Wald's theory of general relativity, where they represent the change in a quantity along a specific direction in spacetime. They are also used in economic and financial models to analyze the impact of changes in specific variables on overall outcomes.

Can (n,m) to (n+1,m) derivatives be negative?

Yes, (n,m) to (n+1,m) derivatives can be negative. This indicates that the function is decreasing along the (n+1)th dimension, while keeping all other dimensions constant at m.

How are (n,m) to (n+1,m) derivatives used in practical applications?

(n,m) to (n+1,m) derivatives are used in a variety of fields, including physics, economics, and engineering. They can help to analyze the impact of changes in specific variables on overall outcomes, and are also useful in optimization and control problems.

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