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Equation of a level surface of a function with 3 variables

  1. Aug 28, 2011 #1
    1. The problem statement, all variables and given/known data
    G6QH5.jpg
    I'm having problems understanding what F(x,y,z)=k means. What does "it is a level surface of a function F of three variables" mean? If it's a surface, why not describe it as z=f(x,y)?

    Thanks
     
  2. jcsd
  3. Aug 28, 2011 #2

    LCKurtz

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    You could, in principle, write as z = f(x,y). But you might not be able to solve the equation for z. Anyway, the F(x,y,z) form is more symmetric and sometimes easier to work with. For example, x2+y2+z2=9 is in many ways "nicer" to deal with than z = ±sqrt(9-x2+y2). You will see when you start dealing with gradients, which the above quote looks like it is leading up to.

    I think the easiest way to think of level surfaces in this context is to imagine that F(x,y,z) is the temperature at any point (x,y,z) in a 3-D region. Then F(x,y,z)=k represents the surface where the temperature is k.
     
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