Covariant derivate same as normal derivative?

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Discussion Overview

The discussion revolves around the relationship between covariant derivatives and normal derivatives, specifically in the context of a scalar field within general relativity. Participants explore the application of the Leibniz rule to covariant derivatives and the implications of different types of covariant derivatives.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions whether the covariant derivative of a function like constant * exp(2f) is equivalent to a specific expression involving the covariant derivative of f.
  • Another participant notes that covariant derivatives are defined to satisfy the Leibniz rule and mentions that the formula for a covariant derivative of a composition of functions can be derived from this rule.
  • A participant clarifies that f is a scalar field and expresses confusion about the different types of covariant derivatives, indicating a lack of awareness of variations in their definitions.
  • It is mentioned that covariant derivatives are relevant in the contexts of general relativity and gauge theories.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the equivalence of covariant and normal derivatives, and there is no consensus on the implications of different types of covariant derivatives.

Contextual Notes

There is a lack of clarity regarding the specific definitions and contexts of covariant derivatives being discussed, as well as the implications of the Leibniz rule in these cases.

vitaniarain
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hey, I'm getting really confused with something. If i have the covariant derivative of say, constant * exp(2f), is this equivalent to 2*constant*exp(2f) * cov.deriv. of f ?
 
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As a rule covariant derivatives are defined so as to satisfy the Leibniz rule. The formula for a covariant derivative of a composition of functions can be deduced from this rule.

What is your f? What kind of a covariant derivative? Sometimes, however, it can be a graded Leibniz rule (when you are playing with supersymmetry, for instance). Then one has to be more careful.
 
thanks for the reply. f is a scalar field. what do u mean what kind of covariant derivative i didn't know there were many. this is in the framework of general relativity..
 
vitaniarain said:
thanks for the reply. f is a scalar field. what do u mean what kind of covariant derivative i didn't know there were many. this is in the framework of general relativity..

Covariant derivatives occur in the context of general relativity or in the context of gauge theories.
 

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