How do you integrate eqns with indices?

  • #1
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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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Answers and Replies

  • #2
haushofer
Science Advisor
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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.
 

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