Divergence of a Vector Gradient
Formula
The divergence of the gradient $\nabla \mathbf{u}$ of the vector function $\mathbf{u} : \mathbb{R}^{n} \to \mathbb{R}^{n}$ is given by: $$ \nabla \cdot \nabla \mathbf{u} = \nabla^{2} \mathbf{u} $$
Explanation
This formula is used when computing the divergence of the symmetrized gradient in the derivation of the Navier–Stokes equations.
The point of this post is less the derivation itself than to give a tangible sense of what $\nabla^{2} \mathbf{u}$ actually looks like.
Derivation
$$ \nabla \mathbf{u} = \nabla \left( u_{1} , u_{2} , u_{3} \right) = \begin{bmatrix} {\frac{ \partial u_{1} }{ \partial x_{1} }} & {\frac{ \partial u_{1} }{ \partial x_{2} }} & {\frac{ \partial u_{1} }{ \partial x_{3} }} \\ {\frac{ \partial u_{2} }{ \partial x_{1} }} & {\frac{ \partial u_{2} }{ \partial x_{2} }} & {\frac{ \partial u_{2} }{ \partial x_{3} }} \\ {\frac{ \partial u_{3} }{ \partial x_{1} }} & {\frac{ \partial u_{3} }{ \partial x_{2} }} & {\frac{ \partial u_{3} }{ \partial x_{3} }} \end{bmatrix} = \begin{bmatrix} \partial_{1} u_{1} & \partial_{2} u_{1} & \partial_{3} u_{1} \\ \partial_{1} u_{2} & \partial_{2} u_{2} & \partial_{3} u_{2} \\ \partial_{1} u_{3} & \partial_{2} u_{3} & \partial_{3} u_{3} \end{bmatrix} $$
Without loss of generality, we will perform a brute-force calculation only for the case $n = 3$. Here, $\partial_{i}$ denotes the partial derivative operator with respect to the $i$-th variable. Such a straightforward calculation is often more useful in fluid mechanics and related fields.
$$ \begin{align*} \nabla \cdot \nabla \mathbf{u} =& \nabla \cdot \begin{bmatrix} \partial_{1} u_{1} & \partial_{2} u_{1} & \partial_{3} u_{1} \\ \partial_{1} u_{2} & \partial_{2} u_{2} & \partial_{3} u_{2} \\ \partial_{1} u_{3} & \partial_{2} u_{3} & \partial_{3} u_{3} \end{bmatrix} \\ =& \begin{bmatrix} \partial_{1} \partial_{1} u_{1} + \partial_{2} \partial_{2} u_{1} + \partial_{3} \partial_{3} u_{1} \\ \partial_{1} \partial_{1} u_{2} + \partial_{2} \partial_{2} u_{2} + \partial_{3} \partial_{3} u_{2} \\ \partial_{1} \partial_{1} u_{3} + \partial_{2} \partial_{2} u_{3} + \partial_{3} \partial_{3} u_{3} \end{bmatrix} \\ =& \begin{bmatrix} \nabla^{2} u_{1} \\ \nabla^{2} u_{2} \\ \nabla^{2} u_{3} \end{bmatrix} \\ =& \nabla^{2} \mathbf{u} \end{align*} $$
The reason this is confusing is that the extension of the definition of the Laplacian to vector functions is often omitted, or the convention for reading the matrix may be ambiguous.
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