Verifying Fluid Dynamics Equation

[ICSunSpots]

1. Verify that v_j∂_jv_i=∂_i(1/2*v²)-ε_ijkv_jω_k,where ω_k=ε_klm∂_lv_m

Homework Equations

The Attempt at a Solution

I am confused on how to proceed with this problem up to this point I have decomposed ω_k into Curl(v). Which leaves v_j X Curl(v). Decomposing this leaves v_m∇_iv_m-v_l∇_lv_i. I am stuck on where to go from here. How do I get rid of the ]=∂_i(1/2*v²) term? Any help is appreciated. Thanks.

 

[tiny-tim]

Welcome to PF!

1. Verify that v_j∂_jv_i=∂_i(1/2*v²)-ε_ijkv_jω_k,where ω_k=ε_klm∂_lv_m

Hi ICSunSpots! Welcome to PF!

Hint: ∂_i(1/2*v2) = v_j∂_iv_j, isn't it?

 

[ICSunSpots]

∂_i(1/2*v²) = v_j∂_iv_j


Thank you for your reply. That would certainly make my solution thus far work.
However,
I am fairly new to index notation. Can you explain to me in vector notation what v_j∂_iv_j means? I understand the ]∂_iv_j term, but what does the vector v_j in front of it do? Thanks.

 

[tiny-tim]

∂_i(1/2*v²) = v_j∂_iv_j


Thank you for your reply. That would certainly make my solution thus far work.
However,
I am fairly new to index notation. Can you explain to me in vector notation what v_j∂_iv_j means? I understand the ]∂_iv_j term, but what does the vector v_j in front of it do? Thanks.

I'm using the summation convention … repeated indices are added over all possible basis values.

So v² = v_jv_j,

and so ∂_iv² = ∂_i(v_jv_j) = (∂_iv_j)v_j + v_j(∂_iv_j) = 2v_j∂_iv_j

 

[ICSunSpots]

Great! It's clear as day now. This index notation is really neat stuff, it just takes a little bit for it to be intuitive. Thank you for your help, I can now finish the problem.