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Find a function of two variables with these propertes . . .

  1. Oct 4, 2009 #1
    1. The problem statement, all variables and given/known data
    Find a function of two variables whose level curves are parabolas with vertex (0,0) with a hole in the parabolas at the origin.


    2. Relevant equations
    No special equations come to mind.
    4ay=x2 may be a little useful.


    3. The attempt at a solution

    The first thing I thought of was y=x3/x, but that doesn't include the third variable. I know the parabolas can have any value for a or face any direction, but I do not know how to include z so that the function will remain having level curves as stated.

    Are there any other functions with these properties? Any input will be appreciated.
     
  2. jcsd
  3. Oct 4, 2009 #2

    lanedance

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    Homework Helper

    so i assume by z you mean the function z = F(x,y) we are trying to find?

    so we have the level curves of z = F(x,y), given by F(x,y) = c, for some constant a, are given by y = ax^2, for some constant a.

    What if we set the constants the same?

    F(x,y) = a --> y = a^x2
    hopefully this will get you started...

    a few other things to consider.
    - The solution for F(x,y) is not uniquie - can you find the full family of solutions?
    - The gradient of F(x,y) will be perpindicular to a level curve at any point, useful check
    - The levels curevs have a "hole" at (0,0), why & what does this imply for F(x,y)?
     
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