What is the meaning of the notation \partial \betaD\alpha in General Relativity?

In summary, the notation \partial \betaD\alpha refers to the set of all partial derivatives of a vector function and can also represent the "outer derivative" of a vector in the context of General Relativity.
  • #1
Reedeegi
99
0
What does the notation [tex]\partial[/tex] [tex]\beta[/tex]D[tex]\alpha[/tex]
mean? I came across it in General Relativity, so I think it's the set of all partial derivatives of the vector function, i.e.
[tex]\partial[/tex]0D1, [tex]\partial[/tex]0D[tex]2[/tex]

and so on... but I'm not entirely sure.
 
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  • #2
If you mean [itex]\partial_{\beta}D^{\alpha}[/itex], then it is short hand for [tex]\frac{\partial D^{\alpha}}{\partial x^{\beta}}[/tex]
 
  • #3
Depending on the context, it can also mean boundary. What is the context?
 
  • #4
Since Reedeegi said it was from General Relativity I suspect the context is as cristo assumed and it is the "outer derivative" of the vector- the tensor represented by array in which the "[itex]\alpha\beta[/itex]" component is the derivative of the [itex]\alpha[/itiex] component of D with respect to [itex]x^\beta[/itex]- which is, of course, just what cristo said.
 

FAQ: What is the meaning of the notation \partial \betaD\alpha in General Relativity?

What is the partial derivative notation?

The partial derivative notation is a mathematical notation used to represent the partial derivative of a function with respect to one of its variables. It is typically written as ∂f/∂x, where ∂ represents the partial derivative operator, f is the function, and x is the variable.

How is the partial derivative notation different from regular derivative notation?

The partial derivative notation is different from regular derivative notation in that it represents the change in a function with respect to a specific variable while holding all other variables constant. Regular derivative notation, on the other hand, represents the total change in a function.

Why is the partial derivative notation important?

The partial derivative notation is important because it allows scientists and mathematicians to analyze how a function changes with respect to a specific variable while keeping all other variables constant. This is crucial in many fields such as physics, economics, and engineering.

How is the partial derivative notation used in real-world applications?

The partial derivative notation is used in many real-world applications, such as calculating the rate of change of temperature in a room with respect to time, or determining the sensitivity of a stock's price to changes in interest rates. It is also used in multivariable calculus to optimize functions with multiple variables.

Are there any rules or properties associated with the partial derivative notation?

Yes, there are several rules and properties associated with the partial derivative notation, such as the product rule, quotient rule, and chain rule. These rules help simplify the process of finding partial derivatives and are essential in solving more complex problems.

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