Two vector operations and simple expressions

[JiWang]

TL;DR Summary: My problems comes to a vector expression which needs to be simplified

I got an expression
p_i=ε_ijks_k,lu_l,j
Here s and u are two vectors. What will be the vector expression of this vector p with curl s, curl u, and other operations?

 

[BvU]

My problem

Hello @JiWang ,

$\qquad$ !​

Please post the complete problem statement; see https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/

Also, I don't see any vectors on the righthand side in p_i=ε_ijks_klu_lj

$\$

 

[Ibix]

Hello @JiWang ,

$\qquad$ !​

Please post the complete problem statement; see https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/

Also, I don't see any vectors on the righthand side in p_i=ε_ijks_klu_lj

$\$

I thought that too, but on closer inspection there are commas between the subscripts in the OP. Much clearer in LaTeX: $s_{k,l}=\partial_ls_k$ and similarly $u$. @JiWang, you should also read the LaTeX Guide.

 

[JiWang]

Hello @JiWang ,

$\qquad$ !​

Please post the complete problem statement; see https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/

Also, I don't see any vectors on the righthand side in p_i=ε_ijks_klu_lj

$\$

Hi , Thanks.
$p_i$ is the component of a vector in the $\bf_i$ direction.
Also $\epsilon_{ijk}$ is the permutation operator. $u_{i,j}$ is $\frac{du_i}{dx_j}$.

 

[pasmith]

TL;DR Summary: My problems comes to a vector expression which needs to be simplified

I got an expression
p_i=ε_ijks_k,lu_l,j
Here s and u are two vectors. What will be the vector expression of this vector p with curl s, curl u, and other operations?

Use $$(\partial_j u_l) (\partial_l s_k) = \partial_j ( u_l \partial_l s_k) - u_l \partial_j \partial_l s_k.$$ Then you can write your expression as $$\mathbf{p} = \nabla \times (\mathbf{u} \cdot \nabla \mathbf{s}) - \mathbf{u} \cdot \nabla ( \nabla \times \mathbf{s}).$$

 

[JiWang]

Can you expand them to single operations? Thanks.

 

[PeroK]

Can you expand them to single operations? Thanks.

See the homework guidelines. @pasmith has already done more than enough of your homework!