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How to parameterize these surfaces?

  1. Dec 13, 2016 #1
    1. The problem statement, all variables and given/known data
    Calculate ##\iint { y+{ z }^{ 2 }ds } ## where the surface is the upper part of a hemisphere with radius a centered at the origin with ##x\ge 0##

    2. Relevant equations
    Parameterizations:
    ##\sigma =\left< asin\phi cos\theta ,asin\phi sin\theta ,acos\phi \right> ,0\le \phi \le \frac { \pi }{ 2 } ,\frac { -\pi }{ 2 } \le \theta \le \frac { \pi }{ 2 } \\ N=(asin\phi )\sigma \\ \left| N \right| ={ a }^{ 2 }sin\phi \\ \\ \alpha =\left< rcos\theta ,rsin\theta ,0 \right> ,0\le r\le a,\frac { -\pi }{ 2 } \le \theta \le \frac { \pi }{ 2 } \\ N=-k\\ \left| N \right| =1##

    3. The attempt at a solution
    are these the right parameterizations?
     
  2. jcsd
  3. Dec 14, 2016 #2

    BvU

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    Check for yourself: ##\ \sigma =\left< a\sin\phi \cos\theta ,a\sin\phi \sin\theta ,a\cos\phi \right> \ ##seems right to me. For ##\ \iint ds\ ## you would then get ##\ \pi a^2, \ ## right ?

    It is not clear to me what you do to express ##\ ds \ ##. What is ##N## and what is the function of ##N## ?
     
  4. Dec 14, 2016 #3
    Oh sorry, by the integral I mean a surface integral. N is the normal. Both parameterizations seem right to me...i originally had ##a## instead of ##r## for the second parameterization. But that would just give me a circle, not a disk (a surface)
     
  5. Dec 14, 2016 #4

    BvU

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    Are we mixing up two threads with almost the same title ?
    Not clear to me why you need ##N## in this thread. But you sure need ##ds## and I haven't seen how you are going to express that in the parameters
     
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