Equation of a level surface of a function with 3 variables

[davidp92]

Homework Statement

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

 

[LCKurtz]

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.