Graphing the Surface y^2 + z^2 = 1 in Cylindrical Coordinates

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Homework Help Overview

The discussion revolves around the identification and sketching of the surface defined by the equation y² + z² = 1, specifically in the context of cylindrical coordinates. Participants are tasked with showing at least one contour perpendicular to each coordinate axis.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants attempt to analyze the surface by examining its projections in the yz, xy, and xz planes. There is confusion regarding the nature of the surface and the implications of the absence of an x variable in the equation. Questions arise about the meaning of "straight lines" in the context of the xy and xz planes, and the need for additional cross sections to better understand the 3D model is suggested.

Discussion Status

The discussion is ongoing, with participants exploring various interpretations of the surface and seeking clarification on the implications of the equation. Some guidance has been provided regarding the need for more cross sections to visualize the surface effectively, but there is no explicit consensus on the final interpretation yet.

Contextual Notes

Participants note the lack of an x variable in the equation, leading to discussions about the arbitrary nature of x values and the dimensionality of the problem. The original poster expresses uncertainty about the 3D model, indicating a need for further exploration of the surface's characteristics.

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Homework Statement


Identify and sketch the graph of the surface y^2 + z^2 = 1. Show atleast one contour perpendicular to each coordinate axis

Homework Equations





The Attempt at a Solution



for the yz plane z = (1-y^2)^1/2 a circle of radius 2 centered at the origin

xy, set z=0 yields y = +/- (1)^1/2 which is a straight line at y+ and y-

xz, set y = 0 yields z = +/- 1^1/2 which is a straight line at z+ & z-


i do not get what the 3d model should look like
 
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joemama69 said:

Homework Statement


Identify and sketch the graph of the surface y^2 + z^2 = 1. Show atleast one contour perpendicular to each coordinate axis

Homework Equations





The Attempt at a Solution



for the yz plane z = (1-y^2)^1/2 a circle of radius 2 centered at the origin
No. For the y-z plane (IOW, x = 0), you have y2 + z2 = 1, which is a circle of radius 1, centered at the origin.

Solving for z was really a wasted effort, since this is the equation of a circle.

joemama69 said:
xy, set z=0 yields y = +/- (1)^1/2 which is a straight line at y+ and y-
What does "straight line at y+ and y-" mean? In the x-y plane you have y = +/- 1, two horizontal lines
joemama69 said:
xz, set y = 0 yields z = +/- 1^1/2 which is a straight line at z+ & z-
Same comment as above.
joemama69 said:
i do not get what the 3d model should look like

What you need are more cross sections, such as the cross section in the planes x = 1, x = 5, x = 10, x = -1, x = -5, and x = -10. After you understand why I picked these cross sections, and graph a few of them, you'll have a good idea of what this surface looks like.
 
but there is no x value in the function.
 
joemama69 said:
but there is no x value in the function.
Which means that it is arbitrary; you can choose any value for x.

A clue that you're dealing in three dimensions is that they asked for the surface. If the problem had asked you to graph the curve y2 + z2= 1, you would have needed only two dimensions.
 

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