Proof of Vector Field Identity: (u.∇)u = ∇(1/2u^2)+w∧u

[connor415]

u is a vector field,

show that

(u.∇)u = ∇(1/2u^2)+w∧u

Where w=∇∧u

 

[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.

 

[connor415]

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

 

[connor415]

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

 

[connor415]

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

 

[gabbagabbahey]

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

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 $\mathbf{\nabla}\left(\frac{1}{2}u^2\right)$?

 

[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.

 

[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.

 

[latentcorpse]

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

 

[connor415]

no magic

 

[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...

 

[connor415]

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

 

[latentcorpse]

have u seen Levi Civita symbols before?

 

[Redbelly98]

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?