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Representing a region as limits of a volume integral

  1. Aug 28, 2011 #1
    1. The problem statement, all variables and given/known data

    i have the region given as being bounded by x2+y2=4 and z=0 and z=3.
    this problem asks to prove gauss divergence theorem for a given vector F

    2. Relevant equations
    3. The attempt at a solution
    As for the volume integral, i had no problem. But for the surface integral, how many surfaces are there actually? How are the normals represented?


    I assumed it is a cylinder(Is it??).. and i got the normal vectors to the top surface and the bottom surfaces to be k and -k respectively. But these cancel out effectively. The problem here is i dont know how to represent any other surface, if any.. i really do not know to find the others. could you give me an idea on how to go about this ? i mean, finding the surfaces and limits..
     
  2. jcsd
  3. Aug 28, 2011 #2

    LCKurtz

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    Yes, the side surface is a cylinder. You might parameterize the surface x2+y2 = 4 for z between 0 and 3 like this:

    R(θ,z) = <2cos(θ), 2sin(θ),z>, 0 ≤θ≤ 2pi, 0≤z≤3
     
  4. Aug 28, 2011 #3

    HallsofIvy

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    There are three "sides", the cylindrical [itex]x^2+ y^2= 4[/itex] and two circular ends at z= 0 and z= 3.
    For those good parametric equations would be [itex]x= r cos(\theta)[/itex], [itex]y= r sin(\thet)[/itex], with r from 0 to 2 and [itex]\theta[/itex] from 0 to [itex]2\pi[/itex], z= 0 or 3.
     
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