Divergence as the Trace of a Jacobian Matrix
If\[ \mathbf{F}=F_1(x_1 , x_2 ,...,x_n) \mathbf{e_1} +...F_n(x_1 , x_2 ,...,x_n) \mathbf{e_n}\]
is a vector field with \[n\]
arguments \[x_1 , ...,x_n\]
and \[n\]
components \[F_1 , ..., F_n\]
then\[\begin{equation} \begin{aligned} div \mathbf{F} &= \mathbf{\nabla} \cdot \mathbf{F} \\ &=( \frac{\partial}{\partial x_1} \mathbf{e_1} +... + \frac{\partial}{\partial x_b} \mathbf{e_n}) \cdot (F_1(x_1 , x_2 ,...,x_n) \mathbf{e_1} +...+ F_n(x_1 , x_2 ,...,x_n) \mathbf{e_n}) \\ &= \sum_{i=1}^n \frac{ \partial F_i}{\partial x_i} \\ &= Tr \frac{\partial (F_1 ,...,F_n}{\partial (x_1 ,..., x_n)} \end{aligned} \end{equation}\]