The author of this textbook is introducing this coordinate free definition of the divergence of tensor fields. Can someone help break this down for me, I am just really confused. First of all, does "a" represent a vector field? And then the the dot product of the two vector fields given by <div(T), a> would give us a scalar field right? Also, the right hand side of this equation would represent a scalar field as well. Any help would be greatly appreciated!
2026-04-14 01:42:54.1776130974
Divergence of Tensor Field
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$ \def\r#1{\color{red}{#1}} \def\LR#1{\left(#1\right)} \def\op#1{\operatorname{#1}} \def\p{\partial_i} \def\c{\cdot} \def\n{\nabla} \def\a{a_k} \def\T{T_{ik}} $You will have a hard time proving the linked formula because it is wrong.
To see this, expand $\n\c\LR{T\c a}$ using the Einstein summation convention $$\eqalign{ \p\LR{\T\a} &= \LR{\p\T}\a + \T\LR{\p\a} \\ }$$ then rearrange the terms $$\eqalign{ \LR{\p\T}\a &= \p\LR{\T\a} - \T\LR{\p\a} \\ }$$ and convert back to vector notation $$\eqalign{ \LR{\n\c T}\c a &= \n\c\LR{\r T\c a} - T:\LR{\n a} \\ }$$ or in terms of named operators $$\eqalign{ {\op{div}\LR{T}}\c a &= \op{div}\LR{\r T\c a} - \op{trace}\LR{T^T\!\c{\n a}} \\ }$$ Note that the term in red is $\r{T^T}$ in the linked formula, which is incorrect.
$\sf NB\!:\:$ I could have expanded $\n\c\LR{T^T\c a}\,$ but then the linked document would have two typos instead of one, so the above is actually the most generous interpretation.