# (u.∇)u = ∇(1/2u^2)+w∧u

1. Dec 2, 2009

### connor415

u is a vector field,

show that

(u.∇)u = ∇(1/2u^2)+w∧u

Where w=∇∧u

2. Dec 2, 2009

### latentcorpse

expand both sides

RHS $\frac{1}{2} \partial_i u_j u_j - \epsilon_{ijk} w_k u_j$
$=\frac{1}{2} \partial_i u_j^2 - \epsilon_{ijk} \epsilon_{klm} u_j \partial_l u_m$
$=\frac{1}{2} 2 u_j \partial_i u_j - \left( \delta_{il} \delta_{jm} - \delta_{jl} \delta_{im} \right) u_j \partial_l u_m$
$=u_j \partial_i u_j - \left( u_j \partial_i u_j - u_j \partial_j u_i \right)$

so that side should be fairly easy to finish off and then just expand the LHS in index notation and show they match up and you're done.

Last edited: Dec 2, 2009
3. Dec 2, 2009

### connor415

Sorry Im still confused. How did you expand w∧u?

4. Dec 2, 2009

### connor415

ps I tried to do it starting from the left, could you do it that way please? Thanks

5. Dec 2, 2009

### connor415

oh and were j and m supposed to be upper case in line 3 of your method?

6. Dec 2, 2009

### gabbagabbahey

As per forum rules, you shouldn't be asking latentcorpse to do your homework for you....you need to make an effort yourself.

What is $\mathbf{\nabla}\wedge\textbf{u}$ in index notation?...How about $\mathbf{\nabla}\left(\frac{1}{2}u^2\right)$?

7. Dec 2, 2009

### latentcorpse

those indices were meant to be subscript, sorry.

it will be easiest to expand the LHS and the RHS seperately and then show that the two expansions are easiest rather than trying to expand one side and rearrange it to give the other side.

8. Dec 3, 2009

### connor415

Im not asking him to do my homework. I did it myself. Just his method was different to mine so was asking him to do it same way.

9. Dec 3, 2009

### latentcorpse

how did u do it then? using indices as well, surely?

10. Dec 3, 2009

### connor415

no magic

11. Dec 3, 2009

### latentcorpse

well in answer to your earlier question about the expansion of $w \wedge u$

$(w \wedge u)_i = \epsilon_{ijk} w_j u_k = - \epsilon_{ijk} u_j w_k$

i used the antisymmetry of the Levi Civita in order to have the k index on the w. just because it's easier to expand the w that way...

12. Dec 3, 2009

### connor415

Yeah me too! Nah youve lost me sorry. Cheers for the effort nonetheless

13. Dec 3, 2009

### latentcorpse

have u seen Levi Civita symbols before?

14. Dec 3, 2009

### Redbelly98

Staff Emeritus
connor415,

People use different notation for vectors. Can you show us how you would expand the dot product of two vectors, u.v?

I.e.,

u.v = ux*vx + uy*vy + uz*vz​

or
u.v = ui*vi + uj*vj + uk*vk​

or something else?

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