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Math Help - Parameterisation variables

  1. #1
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    Parameterisation variables

    Let S be the surface of the solid region formed from the intersection of the solid region underneath the cone z = 2 - 3((x^2 + y^2))^(0.5) and above the region above the plane z = 0.

    I suppose I use cylindrical coordinates to parameterise.

    What would be the value of p (i.e rho)? Is it 1/3? And what is the domain of the parameters?

    And are there 3 outward normals?

    Someone please tell me if I am right or wrong. Thanks.

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  2. #2
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    Quote Originally Posted by maibs89 View Post
    Let S be the surface of the solid region formed from the intersection of the solid region underneath the cone z = 2 - 3((x^2 + y^2))^(0.5) and above the region above the plane z = 0.







    I suppose I use cylindrical coordinates to parameterise.

    What would be the value of p (i.e rho)? Is it 1/3? And what is the domain of the parameters?

    And are there 3 outward normals?

    Someone please tell me if I am right or wrong. Thanks.
    Greetings maibs89

    Changing to polar coordinates we get (remember: x^2+y^2=r^2)

     z=2-3r

    setting z=0 gives

    0=2-3r \iff 3r=2 \iff r=\frac{2}{3}

    This tells us how far the radius goes out in the xy plane (z=0)
    So the paramters are

    \theta \in [0,2\pi] \mbox{ and } r \in [0,\frac{2}{3}]

    To find the normal vector we need to take the gradient of F(x,y,z)=0

    z=2-3\sqrt{x^2+y^2} \iff 3\sqrt{x^2+y^2}+z-2=0=F(x,y,z)

    \vec n = \nabla F = \frac{3x}{\sqrt{x^2+y^2}}\vec i +\frac{3y}{\sqrt{x^2+y^2}} \vec j +\vec k

    Note: there is not a normal vector at the tip of the cone x=y=0

    I hope this helps.

    Good luck.
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  3. #3
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    Hmmm,

    This is a bit puzzling.

    Isn't the required region a cylinder? I want to use cylindrical coords, if that's possible.
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  4. #4
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    Quote Originally Posted by maibs89 View Post
    Hmmm,

    This is a bit puzzling.

    Isn't the required region a cylinder? I want to use cylindrical coords, if that's possible.
    Hello maibs89,

    z=2-3r IS in cylindrical coordinates.

    r goes from zero to 2/3 theta from 0 to 2 Pi as I stated before.

    The graph is a cone. Here are a few graphs to help you see it.

    First in rectangular coordinates with the plane z=0 for clarity.

    Parameterisation variables-capture.jpg

    Now just the cone plotted in cylindrical coordinates.

    Parameterisation variables-capture1.jpg

    As for the normal vector to the cone it is the one given in my first post.

    The normal to the bottom of the cone(z=0) plane with be - \vec k

    I Hope this clears it up.

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  5. #5
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    Ok...

    But isnt't the cone shifted upwards by 2?

    I could be wrong on this.
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