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

 

[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!