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Math Help - Parametric represetation of an ellipsoid

  1. #1
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    Parametric represetation of an ellipsoid

    Find the parametric representation of the ellipsoid 9x^2+3y^2+4z^2=16 using the Euler's angles. Then calculate the normal vector and the first quatratic form at (1,1,1).

    Solution so far:

    I use the method I learned in class and got X( \phi , \theta ) = ( \frac {4}{3} sin \theta cos \phi , \frac {4}{ \sqrt {3}} sin \theta sin \phi , 2cos \theta )

    Now, the normal vector n = \frac {X_{ \phi } \times X_{ \theta}} {| X_{ \phi } \times X_{ \theta }|}

    But I found that  X_{ \phi } \times X_{ \theta} = (- \frac {8}{ \sqrt {3}} sin^2 \theta cos \phi , \frac {8}{3} sin^2 \theta sin \phi , - \frac {32}{3 \sqrt {3}} sin \phi sin \theta cos \phi cos \theta ), in which is getting a bit ridiculous, am I doing the right thing?
    Last edited by tttcomrader; April 19th 2008 at 09:31 PM.
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  2. #2
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    The z-component of your cross-product is incorrect:
    \mathbf{X}(\phi,\theta)=(\frac{4}{3}\sin\theta\cos  \phi,\frac{4}{\sqrt{3}}\sin\theta\sin\phi,2\cos\th  eta)
    \mathbf{X}_{\phi}=(\frac{4}{3}\sin\theta\sin\phi,\  frac{4}{\sqrt{3}}\sin\theta\cos\phi,0)
    \mathbf{X}_{\theta}=(\frac{4}{3}\cos\theta\cos\phi  ,\frac{4}{\sqrt{3}}\cos\theta\sin\phi,-2\sin\theta)
    And so
    \mathbf{X}_{\phi}\times\mathbf{X}_{\theta}=\, \,(-\frac{8}{\sqrt{3}}\sin^2\theta\cos\phi,-\frac{8}{3}\sin^2\theta\sin\phi,{\color{red}-\frac{16}{3\sqrt{3}}\sin\theta\cos\theta\sin^2\phi-\frac{16}{3\sqrt{3}}\sin\theta\cos\theta\cos^2\phi  })
    \mathbf{X}_{\phi}\times\mathbf{X}_{\theta}=(-\frac{8}{\sqrt{3}}\sin^2\theta\cos\phi,-\frac{8}{3}\sin^2\theta\sin\phi,{\color{red}-\frac{16}{3\sqrt{3}}\sin\theta\cos\theta})

    That should prove much simpler, particularly when you take the norm.

    --Kevin C.
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