Notation question: partial derivative with arrow

In summary, the conversation discusses the use of the gradient operator in Brown's QFT and how it is denoted with an arrow pointing left or right. It is explained that this direction can be changed by integration by parts and how it translates into conventional notation. The conversation also mentions the use of double arrows to denote a derivative in both directions.
  • #1
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I'm reading thru Brown's QFT.

He uses the notation of the gradient operator or a partial differentiation operater with an arrow over the operator. The arrow points either left or right.

Can someone please tell me what this means?

thanks
 
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  • #2
It just denotes if the derivative is acting to the left or the right. You can usually change this direction by integration by parts.

if the arrow is point to the left and u had [tex] f\overleftarrow{\nabla} g[/tex] it really just means [tex] (\nabla f) g [/tex].

So integrating:

[tex] \int d^4x f\overleftarrow{\nabla} g= [fg]-\int d^4x f\overrightarrow{\nabla} g[/tex], providing functions go to zero at infinity (compact support), you just throw away the [] term.

Translating into conventional notation:

[tex] \int d^4x (\nabla f) g= [fg]-\int d^4x f(\nabla g)[/tex]
 
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  • #3
Two more things to add:

1. The reason one wants to do this is that when using the Feynman "slash" notation, the derivative operator also carries a gamma matrix, which can't be moved to the other side of the field being differentiated.

2. A derivative with a double arrow on top means:

[tex]\alpha \overleftrightarrow {\partial_\mu} \beta = (\partial_\mu \alpha) \beta - \alpha \partial_\mu \beta[/tex]

But be careful, because sometimes this is defined to mean the other way around!
 
  • #4
Thank you very much for the clarification.
 

1. What is a partial derivative?

A partial derivative is a mathematical concept used to describe how a function changes with respect to one of its variables while holding the other variables constant. It is represented by the symbol ∂ and is commonly used in multivariate calculus.

2. How is a partial derivative different from a regular derivative?

A partial derivative is different from a regular derivative because it only considers the change of a function with respect to one variable, while a regular derivative takes into account the change with respect to all variables. This allows for the analysis of how a function changes in a specific direction.

3. What does the arrow above the partial derivative symbol mean?

The arrow above the partial derivative symbol (∂) indicates the variable with respect to which the derivative is being taken. This is important because it helps to clarify which variable is being held constant in the calculation.

4. How is a partial derivative calculated?

A partial derivative is calculated by treating all other variables as constants and using the standard rules of differentiation. For example, to find the partial derivative of a function f(x,y) with respect to x, we would hold y constant and differentiate with respect to x.

5. What are some real-world applications of partial derivatives?

Partial derivatives have many practical applications in fields such as physics, economics, and engineering. They are used to analyze rates of change in complex systems and to optimize functions with multiple variables. For example, they can be used to determine the maximum or minimum values of a function in order to make the most efficient use of resources.

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