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Finding functions that are a level set and a graph

  1. Sep 23, 2016 #1
    1. The problem statement, all variables and given/known data
    This problem concerns the surface determined by the graph of the equation ##x^2 + xy -xz = 2##

    a) Find a function ##F(x,y,z)## of three variables so that this surface may be considered to be a level set of F.

    b) Find a function ##f(x,y)## of two variables so that this surface may be considered to be the graph of ##z=f(x,y)##.

    2. Relevant equations


    3. The attempt at a solution
    I'm not very sure about what I'm saying, but it was nearing my professor's office hours when I asked him this question. He replied something like functions of a level set needs to have three variables, while functions of a graph needs an extra variable, and this applies to all cases, not just the question I'm referring to. So, following his example, I answered (a) like this:

    Let ##F(x,y,z)## be the set ## [x,y,z,w | w=F(x,y,z)] ##, and ##F(x,y,z) = x^2 + xy -xz##, because I was thinking that since this equation has three variables, it's a function of ##\mathbb {R}##3##→ \mathbb {R}##4, so the level curves will have three variables. I don't really know what to do with the constant 2 in the equation though. I think putting the constant 2 there is like setting the value of ##w## to find a level curve. I'm not sure I'm right.


    For (b), I just answered ##z= x + y - \frac{2}{x}##

    Did I answer these correctly?

    Thanks.
     
  2. jcsd
  3. Sep 23, 2016 #2

    andrewkirk

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    Yes your answers are correct. The 2 can be ignored because it is a constant, so it doesn't change what the level sets are, only the value that is obtained on each level set. In fact any function of the form ##x^2+xy-xz+C## for constant ##C## is a correct answer to (a). The simplest such function is the one you gave, which has ##C=0##.
     
  4. Sep 23, 2016 #3
    Thanks!
     
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