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Intermediate step of Derivation

I’m trying to understand the derivation of the Maxwell stress tensor in Heald and Marion. Im confused how they go from 4.101 to 4.102 in the image above. I can't seem to see how 4.101 is the divergence of the tensor described in 4.102. I think my misunderstanding might stem from the fact that I'm not sure how to take the divergence of a tensor. Can someone explain how this is done / how one goes between the two equations in the image?

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In index notation, the divergence of a vector is $\partial_iA_i$ and by analogy the divergence of a tensor with two indices means either $\partial_iA_{ij}$ or $\partial_jA_{ij}$. In the case of a symmetric tensor, these are the same thing.

Note that taking the divergence of a tensor with two indices produces a vector, while taking the divergence of a vector produces a scalar.

When working with Cartesian components in 3D space, you can keep all indices lowered (or raised). A repeated (“contracted”) index implies a sum over the index values 1, 2, 3 for $x, y, z$.

With this information, you should be able to take the divergence of $T_{ij}$. To compare it with 4.101, express 4.101 in index notation as well. If you don’t understand double cross products in index notation, make that a separate question.

I don’t want to provide too complete a solution because working this out is sometimes a homework problem.

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