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- TL;DR Summary
- Square of an integral containing a Green's Function.

Let's say you have a tensor u with the following components:

$$u_{ij}=\nabla_i\nabla_j\int_{r'}G(r,r')g(r')dr'$$

Where G is a Green function, and g is just a normal well behaved function. My question is what is the square of this component? is it

$${u_{ij}}^2=\bigg[\nabla_i\nabla_j\int_{r'}G(r,r')g(r')dr'\bigg]\bigg[\nabla_i\nabla_j\int_{r''}G(r,r'')g(r'')dr''\bigg]$$

or is it

$${u_{ij}}^2=\bigg[\nabla_i\nabla_j\int_{r'}G(r,r')g(r')dr'\bigg]\bigg[{\nabla_i}'{\nabla_j}'\int_{r''}G(r',r'')g(r'')dr''\bigg]$$

The first one makes much more sense to me, but I'm trying to reproduce results from a paper and it appears that they have used the second one. Any help is appreciated.

EDIT:

I'm sorry but it appears I have written the second one wrong. What I mean is:

$${u_{ij}}^2=\nabla_i\nabla_j\int_{r'}G(r,r')g(r')\bigg[{\nabla_i}'{\nabla_j}'\int_{r''}G(r',r'')g(r'')dr''\bigg]dr'$$

I understand that r and r'' are dummy variables, but my problem is writing G in the second integral as G(r',r'') and not G(r,r''). To me it should be the latter because after integrating I should have something depending on r and not r'. If it depends on r' then the result of the second integral will contribute to the first integral (which is over r').

$$u_{ij}=\nabla_i\nabla_j\int_{r'}G(r,r')g(r')dr'$$

Where G is a Green function, and g is just a normal well behaved function. My question is what is the square of this component? is it

$${u_{ij}}^2=\bigg[\nabla_i\nabla_j\int_{r'}G(r,r')g(r')dr'\bigg]\bigg[\nabla_i\nabla_j\int_{r''}G(r,r'')g(r'')dr''\bigg]$$

or is it

$${u_{ij}}^2=\bigg[\nabla_i\nabla_j\int_{r'}G(r,r')g(r')dr'\bigg]\bigg[{\nabla_i}'{\nabla_j}'\int_{r''}G(r',r'')g(r'')dr''\bigg]$$

The first one makes much more sense to me, but I'm trying to reproduce results from a paper and it appears that they have used the second one. Any help is appreciated.

EDIT:

I'm sorry but it appears I have written the second one wrong. What I mean is:

$${u_{ij}}^2=\nabla_i\nabla_j\int_{r'}G(r,r')g(r')\bigg[{\nabla_i}'{\nabla_j}'\int_{r''}G(r',r'')g(r'')dr''\bigg]dr'$$

I understand that r and r'' are dummy variables, but my problem is writing G in the second integral as G(r',r'') and not G(r,r''). To me it should be the latter because after integrating I should have something depending on r and not r'. If it depends on r' then the result of the second integral will contribute to the first integral (which is over r').

Last edited: