Draw or describe the level surface and an intersection of the graph for the function $$f: \mathbb{R}^3 \rightarrow \mathbb{R}, (x, y, z) \rightarrow x^2+y^2$$
I have done the following:
The level surfaces are defined by $$\{(x, y, z) \mid x^2+y^2=c\}$$
- For $c=0$ we have that $x^2+y^2=0$. So for $c=0$, the level set consists of the $z-$axis.
- For $c<0$, the level set is the empty set.
For $c>0$, the level set is the cylinder $x^2+y^2=c$.
Is this correct??
Could I improve something??
How can we describe an intersection??
This looks good so far. Now we need to describe the solutions to $$ 8\,x^2+2\,y^2+18\,z^2=c\tag{1} $$ for $c>0$. Note that we may rewrite (1) as $$ \frac{x^2}{u^2}+\frac{y^2}{v^2}+\frac{z^2}{w^2}=1\tag{2} $$ where \begin{align*} u &= \sqrt{\frac{c}{8}} & v&= \sqrt{\frac{c}{2}} & w&= \sqrt{\frac{c}{18}} \end{align*} The equation (2) is exactly the equation of an ellipsoid. So, in summary, the level sets are