Level Surfaces & Intersection of a Graph: Exploring $f(x,y,z) = x^2+y^2$

[mathmari]

Hey!

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?? (Wondering)

Could I improve something?? (Wondering)

How can we describe an intersection?? (Wondering)

 

[I like Serena]

Hi! (Blush)

Is this correct?? (Wondering)

Yep. (Nod)

Could I improve something?? (Wondering)

Nope. (Shake)

How can we describe an intersection?? (Wondering)

I'm not sure what is intended here. Can you clarify? Or give an example? (Wondering)

As I see it, the cylinder surface is an intersection of the function with the level $c$.
Or perhaps an intersection with a plane is intended, in which case an ellipse will come out (possibly degenerated). (Thinking)

 

[mathmari]

Nope. (Shake)

When I describe the level set at the case when $c>0$ is it enough to say that it is a cylinder or do I have to say also something else for example to mention the radius?? (Wondering)

I'm not sure what is intended here. Can you clarify? Or give an example? (Wondering)

As I see it, the cylinder surface is an intersection of the function with the level $c$.
Or perhaps an intersection with a plane is intended, in which case an ellipse will come out (possibly degenerated). (Thinking)

In my book there is the following definition:

The intersection of the graph of $f$ is the intersection of the graph with a vertical plane.

For example, if we have $f(x, y)=x^2+y^2$ we have the following:

If $P_1$ is the plane $xz$ in $\mathbb{R}^3$ that is defined by $y=0$, then the intersection of $f$ is the set $$P_1 \cap \text{ graph } f=\{(x, y, z) \mid y=0, z=x^2\}$$
that is a parabola in the plane $xz$.
Similarily, if $P_2$ is the plane $yz$, that is defined by $x=0$, then the intersection $$P_2 \cap \text{ graph } f=\{(x, y, z) \mid x=0, z=y^2\}$$ is a parabola in the plane $yz$.
So, do we take which vertical plane we want?? (Wondering)

 

[I like Serena]

It would be nice to mention the radius.

Isn't your example the same as your problem? (Wondering)