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

  • Thread starter connor415
  • Start date
In summary, Connor415, people use different notation for vectors. You can show us how you would expand the dot product of two vectors, u.v.
  • #1
connor415
24
0
u is a vector field,

show that

(u.∇)u = ∇(1/2u^2)+w∧u

Where w=∇∧u
 
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  • #2
expand both sides

RHS [itex]\frac{1}{2} \partial_i u_j u_j - \epsilon_{ijk} w_k u_j[/itex]
[itex]=\frac{1}{2} \partial_i u_j^2 - \epsilon_{ijk} \epsilon_{klm} u_j \partial_l u_m [/itex]
[itex]=\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[/itex]
[itex]=u_j \partial_i u_j - \left( u_j \partial_i u_j - u_j \partial_j u_i \right)[/itex]

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:
  • #3
Sorry I am still confused. How did you expand w∧u?
 
  • #4
ps I tried to do it starting from the left, could you do it that way please? Thanks
 
  • #5
oh and were j and m supposed to be upper case in line 3 of your method?
 
  • #6
connor415 said:
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 [itex]\mathbf{\nabla}\wedge\textbf{u}[/itex] in index notation?...How about [itex]\mathbf{\nabla}\left(\frac{1}{2}u^2\right)[/itex]?
 
  • #7
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
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
how did u do it then? using indices as well, surely?
 
  • #10
no magic
 
  • #11
well in answer to your earlier question about the expansion of [itex] w \wedge u[/itex]

[itex](w \wedge u)_i = \epsilon_{ijk} w_j u_k = - \epsilon_{ijk} u_j w_k[/itex]

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
Yeah me too! Nah youve lost me sorry. Cheers for the effort nonetheless
 
  • #13
have u seen Levi Civita symbols before?
 
  • #14
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?
 

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

1. What is a vector field identity?

A vector field identity is a mathematical equation that relates two vector fields. It shows that the two fields are equivalent or have a similar structure.

2. What does (u.∇)u mean in the proof of this identity?

(u.∇)u (read as "u dot gradient of u") is a notation used to represent the dot product between the vector field u and the gradient operator ∇. In this proof, it is used to show the relationship between the two vector fields.

3. What does ∇(1/2u^2) represent in the equation?

∇(1/2u^2) (read as "gradient of one-half u squared") represents the gradient of the scalar function 1/2u^2. This term is often used in equations involving energy or potential, as it represents the change in energy or potential with respect to position.

4. What is the significance of w∧u in the equation?

w∧u (read as "w wedge u") represents the cross product between the vector fields w and u. In this proof, it is used to show the relationship between the two vector fields and indicate how they are related to each other.

5. How is this vector field identity useful in scientific research?

This vector field identity is useful in many areas of science, including physics, engineering, and mathematics. It can be used to simplify and solve equations involving vector fields, as well as to analyze and understand the relationship between different vector fields. It also has applications in fluid dynamics, electromagnetism, and other fields that involve vector quantities.

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