How Do Divergence Theorem Variants Apply to Vector and Tensor Fields?

[boyboy400]

Homework Statement

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

Homework Equations

The Attempt at a Solution

1- Let T^ij=ε^ijku_k
and ∫T^ij_,jdV=∫T^ijn_jdS
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_iU_j)dV=∫T^ijU_jU_idS
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

 

[lippyka]

1- Using the divergence theorem: ∫Curl u dV=∫div(T)dV=∫Tn.udS where T is a smooth tensor-valued function and n is the outward normal to the boundary S.2- Using the definition of curl and index notation: ∫(∂Tij/∂xiUj)dV=∫TijUjUidS where U is a smooth vector-valued function.3- Using the definition of divergence and index notation:∫div θu dV=∫n.θu udS where θ is a smooth scalar-valued function and n is the outward normal to the boundary S.