Divergence of a rank-2 tensor in Einstein summation

[Niles]

Homework Statement

Hi

When I want to take the divergence of a rank-2 tensor (matrix), then I have to apply the divergence operator to each column. In other words, I get
<br /> \nabla \cdot M = (d_x M_{xx} + d_y M_{yx} + d_zM_{zx}\,\, ,\,\, d_x M_{xy} + d_y M_{yy} + d_zM_{zy}\,\,,\,\, d_x M_{xz} + d_y M_{yz} + d_zM_{zz})<br />
How would I write this vector in Einstein summation? Is it correct that it would be
<br /> \partial_i M_{ij}<br />

 

[1MileCrash]

Homework Statement

Hi

When I want to take the divergence of a rank-2 tensor (matrix), then I have to apply the divergence operator to each column. In other words, I get
<br /> \nabla \cdot M = (d_x M_{xx} + d_y M_{yx} + d_zM_{zx}\,\, ,\,\, d_x M_{xy} + d_y M_{yy} + d_zM_{zy}\,\,,\,\, d_x M_{xz} + d_y M_{yz} + d_zM_{zz})<br />
How would I write this vector in Einstein summation? Is it correct that it would be
<br /> \partial_i M_{ij}<br />

IIRC, yes.

You have three entries, that distinguished by a new j value, since it appears once, and each entry is a summation over i, since it appears twice.