Is this application chain rule correct inside the divergence?

Question

Here $b:\mathbb{R}^d\to \mathbb{R}^d$ and $x\in \mathbb{R}^d$. Assume $b$ is nice enough to take enough derivatives.

Is there a neater way of writing this application of chain rule inside the divergence ? I want to calculate for $T(x)=Ax$ where $A$ is a positive definite matrix $$ \text{div}_x(b\circ T(x)) $$

I get

$$ \text{div}_x(b\circ T(x))=\sum_{i}\sum_{j}\partial_{x_{i}}T_j(x)\partial_{y_j}b_i(y)\Big|_{y=T(x)}=\sum_{i}\sum_{j}T_{j,i}\partial_{y_{j}}b_i(y)\Big|_{y=T(x)}, $$

but can I write this in a neater way using the divergence notation? EDIT : I think I can write the divergence above as Trace$(AJ_b(Ax))$ ?

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Along the same lines what if $b=D\nabla f$ for some $f:\mathbb{R}^d\to \mathbb{R}$, with $D$ some positive definite square matrix?

Answer 1

Let $\mathbf{u}=b(\mathbf{y})$ where $\mathbf{y}=\mathbf{Ax}$. The differential writes $$ d\mathbf{u} = \mathbf{J}_b d\mathbf{y} = \mathbf{J}_b \mathbf{A} d\mathbf{x} $$ using the Jacobian matrix. The divergence simply writes $ \mathrm{tr} \left( \mathbf{J}_b \mathbf{A} \right) $.