Partial derivative is terms of Kronecker delta and the Laplacian operator

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Discussion Overview

The discussion revolves around expressing the term ## T_{ij} = \partial_i \partial_j \phi ## in terms of the Kronecker delta and the Laplacian operator ## \bigtriangleup = \nabla^2 ##. Participants explore the possibility of a general relation or simplification for this expression.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about a potential relation that could express ## T_{ij} ## using the Kronecker delta and the Laplacian operator.
  • Several participants assert that such a simplification cannot be made.
  • Another participant suggests that while it cannot be simplified generally, there may be special cases where it could be possible.
  • A participant requests clarification on the context of the inquiry, indicating that additional information may influence the discussion.

Areas of Agreement / Disagreement

Participants generally agree that a simplification of ## \partial_i \partial_j \phi ## in terms of the Kronecker delta and the Laplacian operator is not possible as a general rule, though some suggest that special cases might exist.

Contextual Notes

The discussion lacks specific context regarding the application or conditions under which the term is being analyzed, which may affect the validity of the claims made.

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TL;DR
How to write partial derivatives in terms of Kronecker delta and the Laplacian operator?
How can the following term:

## T_{ij} = \partial_i \partial_j \phi ##

to be written in terms of Kronecker delta and the Laplacian operator ## \bigtriangleup = \nabla^2 ##?

I mean is there a relation like:

## T_{ij} = \partial_i \partial_j \phi = ?? \delta_{ij} \bigtriangleup \phi.##

But what are ?? term

Any help is appreciated!
 
Last edited:
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It cannot.
 
Orodruin said:
It cannot.
So there is no any way to simplify ## \partial_i \partial_j \phi ## ?
 
Not really no. Not without knowing more. In some special cases, perhaps, but not as a general rule.

What is the context here?
 

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