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Math Help - Please help. Demonstrate the curve

  1. #1
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    Please help. Demonstrate the curve

    This is a problem that I am having trouble figuring out in my Differential Geometry course. Here it is below:

    Demonstrate that the curve
    r(t) = <a(sint)^2, asin(t)cos(t), acos(t)>
    lies on a sphere and that all normal planes pass through the origin.

    I think r(t) is a vector, but my teach really didn't give too much information on it and I am curious. Could someone help me with this problem? Try to show it and detail so that I can understand it, since this is a relatively new topic for me. Thanks.
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  2. #2
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    Quote Originally Posted by Dream View Post
    This is a problem that I am having trouble figuring out in my Differential Geometry course. Here it is below:

    Demonstrate that the curve
    r(t) = <a(sint)^2, asin(t)cos(t), acos(t)>
    lies on a sphere and that all normal planes pass through the origin.

    I think r(t) is a vector, but my teach really didn't give too much information on it and I am curious. Could someone help me with this problem? Try to show it and detail so that I can understand it, since this is a relatively new topic for me. Thanks.
    well the defintion of a sphere is

    x^2+y^2+z^2=r^2

    the vector eqation gives

    x=a\sin^2(t),y=a\sin(t)\cos(t),z=a\cos(t)

    so

    x^2+y^2+z^2=a^2\sin^4(t)+a^2\sin^2(t)\cos^2(t)+\co  s^2(t)=
    factoring gives
    a^2[\sin^2(t)\left[ \sin^2(t)+\cos^2(t)\right] +\cos^2(t)]=a^2[\sin^2(t)[1]+\cos^2(t)]=a^2(1)=a^2

    so this is and eqation of a sphere centered at the origin with radius a

    x^2+y^2+z^2=a^2

    taking the gradient gives the normal vector to the surface

    F(x,y,z)=x^2+y^2+z^2-a^2

    \nabla F=2x \vec i + 2y \vec j +2z \vec k

    I hope this helps. Good luck
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  3. #3
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    Thank you.
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