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Homework Help: A question that uses divergence thm

  1. Apr 18, 2012 #1
    1. The problem statement, all variables and given/known data


    show that the volume enclosed by a closed surface S is given by

    ## \frac{1}{3} \int_{S} \vec{x} \cdot d\vec{A} ##

    2. Relevant equations


    divergence theorem

    3. The attempt at a solution

    using divergence theorem

    I get that ##V =\frac{1}{3} \int_{V} \nabla \cdot \vec{x} dV ##

    but I want

    ## V = \int_{V} dV ## don't I? how does it simplify?
     
  2. jcsd
  3. Apr 18, 2012 #2
    Unless

    ## \vec{x} = (x,y,z) ##

    Then:

    ## V = \frac{1}{3} \int_{V} (1+1+1) dV = \int_{V}dV ##

    It's because x is just the position vector whose divergence is 3 isnt it? i.e. x is not a vector field?
     
  4. Apr 18, 2012 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Sure, that's it. But the position vector (x,y,z) IS a vector field. It's just a particular one.
     
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