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Math Help - regular surface- solution check

  1. #1
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    regular surface- solution check

    For which values of c\in\mathbb{R}, is f^{-1}(c) a regular surface , where
    f:\mathbb{R}^3\to \mathbb{R} is defined as
    f(x,y,z)=x^2y^2+x^2z^2+y^2z^2-4xyz


    I found 2 theorems I think I should use when solving this problem

    thm 1
    c\in f(U) is a regular value of the function f:U\subset\mathbb{R}^3\to \mathbb{R} iff the partial derivatives f_{x},f_{y},f_{z} do not disappear instantly for any point of the pre-image f^{-1}(c)=\{(x,y,z)\in U|f(x,y,z)=c\}


    thm 2


    If f:U\subset\mathbb{R}^3\to \mathbb{R}is differentiable and c\in f(U) is a regular value , then f^{-1}(c) is a regular surface


    So I start off with finding critical points (point where all partial derivatives disappear at the same time)


    \nabla f=(f_{x},f_{y},f_{z})=(2xy^2+2xz^2-4yz,2x^2y+2yz^2-4xz,2x^2z+2y^2z-4xy)


    \nabla f=0 I get the following points:


    c_{1}=(-2,0,0)
    c_{2}=(-1,0,0)
    c_{3}=(-1,1,-1)
    c_{4}=(0,0,0)
    c_{5}=(1,0,0)


    so that means that all point except the above ones are regular points?
    the second theorem gives that f^{-1}(c) is a regular surface \for all c=\{(x,y,z)\in \mathbb{R}^3\setminus \{c_{1},c_{2},c_{3},c_{4},c_{5}\}\}

    Is this solution correct??
    Thank you in advance
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  2. #2
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    Re: regular surface- solution check

    I didn't check your detailed computation but yes the argument is correct.
    Thanks from rayman
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  3. #3
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    Re: regular surface- solution check

    thank you for your reply.
    I would appreciate if you can have a look at my other post where I am trying to show that a function is smooth
    smooth function
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