Divergence of traceless matrix

In summary, the conversation discusses the symmetry of ##\partial M_{ab}/\partial \hat{n}_c## and its relation to the divergence of the traceless part of ##M##. It is stated that the divergence is proportional to the gradient of the trace of ##M##, and the reason for this is explained to be due to the symmetry of ##\partial M_{ab}/\partial \hat{n}_c##.
  • #1
jouvelot
53
2
Assume that ##\partial M_{ab}/\partial \hat{n}_c## is completely symmetric in ##a, b## and ##c##. Then, it is stated in the book I read that the divergence of the traceless part of ##M## is proportional to the gradient of the trace of ##M##. More precisely,
$$ \partial /\partial \hat{n}_a (M_{ab} - \delta_{ab} {\rm Tr} (M)/2) = \partial ({\rm Tr} (M)/2)/\partial \hat{n}_b .$$ Can anyone provide some hints on why this is true, please?
Thanks in advance.
Pierre
 
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  • #2
The trace of [itex]M[/itex] is [itex]M_{aa}[/itex]. By symmetry, [tex]
\frac{\partial M_{ab}}{\partial \hat{n}_a} = \frac{\partial M_{aa}}{\partial \hat{n}_b}
[/tex]
 
  • #3
Thanks. This was pretty simple; I should have gotten that ;)

Bye,

Pierre
 

What is the definition of "Divergence of traceless matrix"?

The divergence of a traceless matrix is a mathematical operation that measures the rate of change of a vector field in a given direction. It is defined as the sum of the partial derivatives of the vector components with respect to the corresponding spatial coordinates.

How is the divergence of a traceless matrix calculated?

The divergence of a traceless matrix is calculated by taking the dot product of the gradient operator and the vector field. This can also be written in index notation as the sum of the partial derivatives of the vector components with respect to the corresponding spatial coordinates.

What is the physical significance of the divergence of a traceless matrix?

The physical significance of the divergence of a traceless matrix is that it represents the net flow of a vector field out of a given point in space. It is used in various fields of physics, such as fluid dynamics and electromagnetism, to describe the behavior of vector fields.

How can the divergence of a traceless matrix be interpreted geometrically?

The divergence of a traceless matrix can be interpreted geometrically as the flux per unit volume of a vector field through an infinitesimal volume element at a given point. It is also related to the concept of divergence as a measure of the spread or convergence of vector field lines.

What is the relationship between the divergence of a traceless matrix and the trace of the matrix?

The divergence of a traceless matrix is always zero, as the traceless property guarantees that the sum of the diagonal elements of the matrix is equal to zero. This is because the trace of a matrix represents the sum of its eigenvalues, and a traceless matrix has eigenvalues that sum to zero.

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