How can i do the divergence of a matrix 3x3?

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    3x3 Divergence Matrix
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Discussion Overview

The discussion revolves around the concept of calculating the divergence of a 3x3 matrix, with participants exploring the definitions and interpretations of divergence in relation to matrices and tensors. The scope includes theoretical considerations and potential applications in fields such as physics and engineering.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant asserts that "divergence" is typically a vector-valued differentiation of a scalar function and is not generally defined for matrices, suggesting that applying divergence to each component could yield a matrix with vector components.
  • Another participant points out that divergence is a scalar quantity and implies that only a 1x1 matrix can represent a scalar.
  • A different viewpoint suggests that the original poster might be referring to the divergence of a vector field in R^3, which could be represented as a 3x3 matrix.
  • Another participant proposes the possibility of interpreting the question as concerning the divergence of a tensor field, which would result in another tensor field of lower valence.
  • The original poster clarifies that they are specifically asking about the divergence of a 3x3 tensor related to the anisotropic diffusion coefficient in 3D, suggesting that it involves taking the derivative of each component with respect to x, y, and z, but expresses uncertainty about this approach.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definition and application of divergence to a 3x3 matrix or tensor. Multiple competing views remain regarding the interpretation of the term "divergence" in this context.

Contextual Notes

There are limitations regarding the assumptions made about the nature of the matrix or tensor in question, as well as the dependence on the variables involved (x, y, z). The discussion also highlights the ambiguity in the interpretation of divergence when applied to matrices or tensors.

germana2006
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How can i do the divergence of a matrix 3x3?
 
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First explain what you are doing! "Divergence" is a vector valued differentiation of a scalar function. It is not, in general, defined on matrices. You could, of course, define it as what you get by applying the divergence to each component of the matrix. The result would be a matrix having vectors as components. How that would be interpreted depends on your original matrix- as a linear operator over a vector space.

And, of course, you haven't said anything about dependence on x, y, z or even how many variables there are. If the matrix were a constant, its diveregence would be "0" although I'm not entirely sure what kind of object that 0 would be!

Could you give more detail about the situation in which you are dealing with the "divergence of a matrix"?
 
Divergence is a scalar. Only a 1x1 matrix can hold a scalar.
 
maybe he means divergence of a vector field on R^3 which could be viewed as a 3x3 matrix. does that make sense? (i am a little over the line at the moment.)
 
Could he mean a "divergence of a tensor field" [which would be another tensor field of lower valence]?
 
Sorry, it's exact the divergence of a Tensor (3x3). I have to do the divergence of the anisotropic Diffusion coefficient in 3D, that's mean a Tensor of 3x3 components.
I think it is: the derivative of each component with respect x, y, and z. But i don't sure.
 

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