Finding functions that are a level set and a graph

[toforfiltum]

Homework Statement

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 $z=f(x,y)$.

Homework Equations

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.

 

[andrewkirk]

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 $C=0$.

 

[toforfiltum]

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 $C=0$.

Thanks!