(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

So I got three things to figure out:

1- ∫Curl u dV=∫u χ n dS

2- ∫div Tu dV=∫T^{T}n. udS

3- ∫div θu dV=∫n.θu udS

where

n defines the outward normal to the boundary S

θ is a smooth scalar-valued function

u is a smooth vector-valued function

T is a smooth tensor-valued function

2. Relevant equations

3. The attempt at a solution

1- Let T^{ij}=ε^{ijk}u_{k}

and ∫T^{ij}_{,j}dV=∫T^{ij}n_{j}dS

Substituting the first one into the integral one (second one) we get the indices form of what we want. So it's solved.

2- ∫(∂T^{ij}/∂x_{i}U_{j})dV=∫T^{ij}U_{j}U_{i}dS

but from here I don't know where to go!

3- I guess if the second one is solved the last one would be easy.

PS. In case these relations have a special name or there is a keyword I can google and find my answers I really appreciate if you can tell me about. Also if there is a book that has the solution please let me know about it. Thank you so much everyone

PS2. Well using the definition of divergence theorem and index notation, I managed to write something...it seems kind of clear but I'm not sure about playing around with the orders and indices especially for the second one where Transpose[T] has to be made at the right hand side like I don't know how to do this ... so hopefully the TA will not be picky this time :D

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# Divergence theory equalities

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