Question on partial derivative

[Samoht]

I just handed in a homework where I used the assumption below

∂_iu_j∂_ju_i=0 ?

but when I start thinking about it I'm not so sure, could someone prove to me that it is zero? Or is that assumption totally off?

Regards

 

[CompuChip]

Is there summation over i and j?
Is u something special, or just an arbitrary vector?

 

[Samoht]

This partial derivative came up in my proof that the rate of strain in fluid dynamics is always positive. u is the velocity and I believe that there is summation but I am very new at this.

The term I tried to prove being positive is:

∂_iu_j((∂_iu_j+∂_ju_i)-2/3 ∂_ku_k δ_ij)

When I multiply ∂_iu_j into the outer bracket then I get

(∂_iu_j(∂_iu_j+∂_ju_i)-2/3 ∂_ku_k δ_ij ∂_iu_j)

which in my mind becomes

(∂_iu_j ∂_iu_j+∂_iu_j ∂_ju_i)-2/3 ∂_ku_k δ_ij ∂_iu_j

the right side of the minus sign can be simplified to 2/3 (∂_ku_k)²

My question relates to the second term in the remaining bracket. Can the term ∂_iu_j ∂_ju_i be set equal to 0? AND, if so why?

Regards