Divergence theory equalities

1. Oct 4, 2011

boyboy400

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=∫TT 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 Tijijkuk
and ∫Tij,jdV=∫TijnjdS
Substituting the first one into the integral one (second one) we get the indices form of what we want. So it's solved.

2- ∫(∂Tij/∂xiUj)dV=∫TijUjUidS
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

Last edited: Oct 5, 2011