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Parametrizing the portion

  1. Oct 22, 2012 #1
    1. The problem statement, all variables and given/known data
    Parametrize the portion of the sphere given by x2 + y2 + z2 = 7 that lies above z = √3. (use cylindrical coordinates)


    2. Relevant equations
    x2 + y2 = r2




    3. The attempt at a solution
    x = rcosθ
    y = rsinθ
    z = z
    z = √3
    x2 + y2 + (√3)2 = 7
    x2 + y2 = 4
    r2cos2θ + r2 sin2θ = 4
    r2 (cos2 + sin2θ) = 4
    r = 2
    Is this the correct way of parametrization?
     
    Last edited: Oct 22, 2012
  2. jcsd
  3. Oct 22, 2012 #2

    SammyS

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    Where is your parametrization ?

    What's your final answer?
     
  4. Oct 22, 2012 #3
    Hmm. I guess my attempt to parametrize was totally incorrect, my question now is how do I parametrize this problem?
     
  5. Oct 22, 2012 #4
    z = √(7 - r2cos2θ - r2sin2θ)
    s(r,θ) = (2cosθ, 2sinθ, √(7 - r2 (cos2θ - sin2θ)) )

    0 < r < 2, 0 < θ < 2∏
     
    Last edited: Oct 22, 2012
  6. Oct 22, 2012 #5

    SammyS

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    You're using the symbol, r, for two different things.

    Are you saying that position vector, (x, y, z), is given by:
    (x, y, z) = (rcosθ, rsinθ, √(7 - r2 (cos2θ - sin2θ)) )​
    ?

    You need to restrict the values of r and θ .
     
  7. Oct 22, 2012 #6
    I apologize for the bad notation, I found that 0 < r < 2 and 0 < θ < 2∏ as restriction
     
  8. Oct 22, 2012 #7
    Ok. let me start from scratch, I'm going to set x = rcosθ , y = rsinθ and z = √(7 - (rcosθ)2 - (rsinθ)2), cylindrical coordinates.

    I found the restriction to r by setting x2 + y2 = r2 in
    portion of sphere given. which gives us r2 + z2 = 7

    Now, it says that portion lies above a plane z = √3.
    I plugged in z, r2 + (√3)2 = 7
    and found that r = 2

    and in cylindrical coordinates the restriction on θ = 2∏

    Next, parametric of the portion, S(x,y) = (x, y, √(7 - x2 - y2))

    Now, I'm stuck here. I'm unsure if this is correct approach in trying to parametrize.
     
    Last edited: Oct 22, 2012
  9. Oct 23, 2012 #8

    SammyS

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    After thinking this over, I believe that your position vector should be of the form: (r, θ, z) .

    Writing x2 + y2 + z2 = 7 in cylindrical coordinates gives: r2 + z2 = 7.

    Solve that for either r or z .
     
  10. Oct 23, 2012 #9
    z = √7 - r2

    s(x,y) = < x, y, (7 - x2 - y2)1/2

    s(r,θ) = < rcosθ, rsinθ, (7 - (rcosθ)2 - (rsinθ)2)1/2
     
  11. Oct 23, 2012 #10

    SammyS

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    What is sin2(θ) + cos2(θ) ?
     
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