# Laplacian of dyad

1. Oct 3, 2016

### ganondorf29

1. The problem statement, all variables and given/known data

Given the dyad formed by two arbitrary position vector fields, u and v, use indicial notation in Cartesian coordinates to prove:

$$\nabla^2 ({\vec u \vec v}) = \vec v \nabla^2 {\vec u} + \vec u \nabla^2 {\vec v} + 2\nabla {\vec u} \cdot {(\nabla \vec v)}^T$$

2. Relevant equations

Per my professor's notes, the Laplacian of a dyad (also a tensor) is given as:
$$\nabla^2 {\mathbf {S}} = \nabla \cdot {S_{ij,k} \mathbf{e_{i}e_{j}e_{k}}} = S_{ij,kk} \mathbf{e_{i}e_{j}}$$

3. The attempt at a solution

$$\nabla^2 {\mathbf {uv}} = (u_{i}v_{j})_{,kk} = u_{i,kk}v_{j} + u_{i}v_{j,kk} \\ u_{i,kk}v_{j} + u_{i}v_{j,kk} = \vec v \nabla^2 {\vec u} + \vec u \nabla^2 {\vec v}$$

I don't know where the following terms come from:
$$2\nabla {\vec u} \cdot {(\nabla \vec v)}^T$$

Does anyone have any suggestions? I feel that I am missing a step or something.

2. Oct 3, 2016

### Staff: Mentor

I'm not good with coordinates, so I leave this up to you.
But $\nabla (\mathbf{u}\mathbf{v}) = \mathbf{u}(\nabla \mathbf{v}) + (\nabla \mathbf{u})\mathbf{v}$ and the next differentiation gives you the third term (twice).

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