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[Multivariable calculus] Find the rate of flow outward through the part of the parabo

  1. Aug 6, 2010 #1
    1. Problem:
    A fluid has density 1000 km/m3 and flows with velocity V =<x,y,z>, where x, y, and z are measured in meters, and the components of V are measured in meters per second. Find the rate of flow outward through the part of the paraboloid z = 64 - x^2 - y^2 that lies above the xy plane.



    2. Relevant Equation:
    Double Integral (Density*V ds)

    Where V is the velocity function




    3. The attempt at a solution
    The only formula i found to solve this problem is formula above(2) which requires me to have the parametric equation for the paraboloid.. The thing is I really have no idea how to turn the paraboloid equation above into parametric equation


    Helps are really appreciated...
     
  2. jcsd
  3. Aug 6, 2010 #2
    Re: [Multivariable calculus] Find the rate of flow outward through the part of the pa

    I don't quite get understand the way you worded the problem, but here's the parametrized surface:

    [tex] \vec{r}(x, \theta) = (xcos\theta, xsin\theta, 64 - x^2) [/tex]

    How is the fluid density measured in km/m^3? What units is your "rate of flow" supposed to be measured in? Also, are you sure your formula is correct?
     
  4. Aug 8, 2010 #3
    Re: [Multivariable calculus] Find the rate of flow outward through the part of the pa

    i just copy pasted the actual problem from wamap.. and about the formula, i'm not sure either since i found it from google...
    anyhow, how do you get the parametric equation? is that something to remember?

    thanks
     
  5. Aug 8, 2010 #4
    Re: [Multivariable calculus] Find the rate of flow outward through the part of the pa

    It's not something you have to memorize, but it is a method with which you should familiarize yourself. I'll give you a hint:

    Fixing z traces out a circle on the paraboloid. The standard and obvious parametrization of the unit circle is [tex] \vec{r}(\theta) = (cos\theta, sin\theta). [/tex] Can you figure out what leads us to the final answer from here?
     
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