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Math Help - Vector Analysis

  1. #1
    Super Member Aryth's Avatar
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    Vector Analysis

    Let f(x,y,z) = x^2 + 2y^2 + 3z^2 and let S be the isotimic surface: f = 1. Find all the points (x,y,z) on S that have tangent planes with normals (1,1,1).

    I'm not even sure where to begin...
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  2. #2
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    We may use the fact that the normal to any point on the level curve of f is parallel to \nabla f at that point. We know this because

    \frac{d}{dt}f(\mathbf{r}(t))=\nabla f(\mathbf{r}(t))\cdot \mathbf{r'}(t)=0

    for a differentiable curve \mathbf{r} on the surface f(x,y,z)=c, which means that \nabla f is perpendicular to the curve and thus to the plane itself.

    In our case, we are to find all those points of f(x,y,z)=1 such that

    \nabla f=\left<c,c,c\right>

    for some constant c.
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