Laplacian of dyad

  • #1

Homework Statement


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

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


Homework Equations


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


The Attempt at a Solution


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


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.
 

Answers and Replies

  • #2
14,884
12,423

Homework Statement


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

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


Homework Equations


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


The Attempt at a Solution


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


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.
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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