Partial derivative is terms of Kronecker delta and the Laplacian operator

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The discussion centers on expressing the term T_{ij} = ∂_i ∂_j φ in terms of the Kronecker delta and the Laplacian operator (∇²). Participants conclude that there is no general simplification of ∂_i ∂_j φ into a form involving the Kronecker delta and the Laplacian without additional context or specific conditions. The consensus is that while special cases may allow for simplifications, a universal relationship does not exist.

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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!
 
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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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