i have a symmetric positive matrix $\boldsymbol{D}(\boldsymbol{x})$ which can be decomposed as:
$\boldsymbol{D}(\boldsymbol{x})$ = $\boldsymbol{B}(\boldsymbol{x}) \cdot\boldsymbol{B}^{T}(\boldsymbol{x})$
if I do apply the divergance to $\boldsymbol{D}(\boldsymbol{x})$ what do i obtain?
(Note that in my notation the divergence is calculated with respect to the second index)
i made some calculus but i wanted to be sure i was correct:
$\nabla \cdot ( \boldsymbol{B}(x) \cdot \boldsymbol{B}^{T}(x) )$ = $\frac{\partial}{\partial x_{i}} B_{jk}(\boldsymbol{x})B_{ki}^{T}(\boldsymbol{x})\boldsymbol{e}_{j}=B_{ki}^{T}(\boldsymbol{x})\frac{\partial}{\partial x_{i}}B_{jk}(\boldsymbol{x}) \boldsymbol{e}_{j}+B_{jk}(\boldsymbol{x})\frac{\partial}{\partial x_{i}}B_{ki}^{T}(\boldsymbol{x}) \boldsymbol{e}_{j}$
which in tensorial notation can be written as:
$\nabla \cdot ( \boldsymbol{B}(x) \cdot \boldsymbol{B}^{T}(x) )=\boldsymbol{B}^{T}(\boldsymbol{x}):(\nabla \boldsymbol{B}^{T}(\boldsymbol{x}))+\boldsymbol{B}(\boldsymbol{x})\cdot(\nabla \cdot \boldsymbol{B}^{T}(\boldsymbol{x}))$
am I correct or am I totally wrong?
This is totally correct.
In your final line, the term "$\boldsymbol{B}^{T}(\boldsymbol{x}):(\nabla \boldsymbol{B}^{T}(\boldsymbol{x}))$" is a bit ambiguous since $(\nabla \boldsymbol{B}^{T}(\boldsymbol{x}))$ has three indices and it's not clear which of them the indices of $\boldsymbol{B}^{T}(\boldsymbol{x})$ are being contracted with.