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

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    3x3 Divergence Matrix
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The discussion centers on the divergence of a 3x3 matrix, specifically in the context of tensor fields. Divergence, typically a scalar operation, is clarified as applicable to each component of a matrix, resulting in a matrix of vector components. The conversation emphasizes that divergence is not inherently defined for matrices but can be interpreted when applied to tensors, particularly in the case of anisotropic diffusion coefficients in three-dimensional space. The participants highlight the need for clarity regarding the variables involved and the nature of the matrix in question.

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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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