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

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Discussion Overview

The discussion revolves around the level surfaces and intersections of the graph for the function $$f(x,y,z) = x^2+y^2$$. Participants explore the definitions and characteristics of level surfaces, particularly for different values of the constant $c$, and seek clarification on how to describe intersections with vertical planes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant describes the level surfaces as defined by $$\{(x, y, z) \mid x^2+y^2=c\}$$ and notes specific cases for $c=0$, $c<0$, and $c>0$.
  • Another participant agrees with the description but questions whether additional details, such as the radius of the cylinder for $c>0$, should be included.
  • There is uncertainty about how to describe intersections, with one participant suggesting that the cylinder surface represents an intersection with the level $c$, while another proposes that intersections with vertical planes yield parabolas.
  • Examples of intersections with specific vertical planes ($P_1$ and $P_2$) are provided, illustrating how these intersections result in parabolic shapes in the respective planes.
  • Participants express a desire for clarification on whether any vertical plane can be chosen for intersection analysis.

Areas of Agreement / Disagreement

Participants generally agree on the basic definitions of level surfaces and intersections, but there is no consensus on the necessity of including additional details such as the radius of the cylinder or the choice of vertical planes for intersections. The discussion remains unresolved regarding these points.

Contextual Notes

Participants express uncertainty about the completeness of their descriptions and the implications of their examples, indicating a need for further exploration of definitions and properties related to level surfaces and intersections.

mathmari
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Hey! :o

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)
 
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Hi! (Blush)

mathmari said:
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)
 
I like Serena said:
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 like Serena said:
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)
 
It would be nice to mention the radius.

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

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