Variable-coefficient Laplacian identity proof 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?

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.
