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Divergence theory equalities

  1. Oct 4, 2011 #1
    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
  2. jcsd
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