Variable-coefficient Laplacian identity proof

Question

Does the following hold?

μ Δu + [(∇u)^T] ∇μ = ∇ · (μ ∇u)

Where:
μ is a scalar function,
u is a vector function,
^T is the transpose operation.

In other words: is variable-coefficient Laplacian combined of const-coefficient Laplacian and product of the transposed vector gradient and coefficient gradient?

Answer 1

Using the product rule to expand the RHS in either index notation $$\eqalign{ \def\BR#1{\left(#1\right)} \def\LR#1{\Big(#1\Big)} \def\a{\mu}\def\b{{\bf v}}\def\n{\nabla}\def\p{\partial} \p_k\LR{\a\:\p_k\b_j} &= \LR{\p_k\a}\LR{\p_k\b_j} + \a\LR{\p_k\p_k\b_j} \\ }$$ or vector notation $$\eqalign{ \p\cdot\LR{\a\n\b} &= \LR{\n\a}\cdot\LR{\n\b} + \a\LR{\n\cdot\n\b} \\ }$$ verifies the relationship.