How do you integrate eqns with indices?

In summary, the conversation discusses an equation involving partial derivatives and its integration. The proposed solution involves functions to be determined by initial/boundary conditions. The conversation also raises questions about the notation and its implications. The analogy to functions of two variables is used to clarify the solution.
  • #1
JuanC97
48
0
Hello, I just want to clarify some things with a simple exercise: I have the equation ## \frac{\partial^2 f}{\partial A^\mu \,\partial A^\nu} = 0## and I want to integrate it once assuming that ## f=f(A^1,A^2,...,A^n)=f(A^\rho) ##.

I think the solution should be ## \frac{\partial f}{\partial A^\mu} = C_1(A^{\rho\neq\mu}) + (1-\delta_{\mu\nu})\,C_2(A^{\rho\neq\nu}) ##
where ##C_1## and ##C_2## are functions that have to be determined by initial/boundary conditions, the first term came from the integral wrt ##A^\nu## when ##\mu=\nu## and the second one when ##\mu\neq\nu##.

The big confusion arises when I think...
1. Does this really make any sense? - It just looks weird.
2. If ## \frac{\partial f}{\partial A^\mu} ## were a covariant tensor, it would make a lot more sense if I had written ## h(A_\mu) ## instead of ## h(A^\mu) ##, wouldn't it?.
 
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  • #2
Compare to e.g. functions of two variables f(x,y). If one has

## \frac{\partial f}{ \partial x} = 0 ##

then the solution is given by

##f(x,y) = f(y)##. I.e. the usual integration constant is promoted to an "integration function of y". So I'd say that if in your case

##\frac{\partial g}{ \partial A^{\nu}} = 0##

we get that the solution is a vector field with lower index which can be any function of all the other fields you have; different fields are indepent per definition.
 

1. How do I determine the correct indices to use in an equation?

To determine the correct indices in an equation, you must first identify the variables and their corresponding indices. Then, consider the mathematical relationship between these variables and how they are affected by each other. This will help you determine the appropriate indices to use in the equation.

2. What is the purpose of integrating equations with indices?

The purpose of integrating equations with indices is to simplify complex mathematical expressions and make them more manageable. By using indices, we can represent multiple variables and their relationships in a concise and organized manner.

3. Can you explain the process of integrating equations with indices?

The process of integrating equations with indices involves identifying the variables and their corresponding indices, determining the appropriate mathematical relationship between these variables, and then performing the necessary operations to integrate the equation. This may include applying the power rule, product rule, or quotient rule, depending on the specific equation.

4. Are there any specific rules or guidelines to follow when integrating equations with indices?

Yes, there are specific rules and guidelines to follow when integrating equations with indices. These include understanding the properties of indices, knowing the basic integration rules, and being familiar with the different types of mathematical relationships that can be represented using indices.

5. Can you provide an example of integrating an equation with indices?

Sure, an example of integrating an equation with indices is integrating the expression x^2y^3 with respect to x. This would involve applying the power rule and the product rule to get the final integral, which would be (1/3)x^3y^3 + C.

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