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

In summary, cylindrical coordinates are a three-dimensional coordinate system used in mathematics and engineering to locate points in space using distance from a central axis, angle from a fixed reference plane, and height or depth from the reference plane. They differ from Cartesian coordinates in that they use a distance, angle, and height instead of three perpendicular axes. The conversion formula between cylindrical and Cartesian coordinates involves using the cosine and sine of the angle to find the x and y coordinates, while the z coordinate remains the same. Cylindrical coordinates are commonly used in real-world applications such as engineering, physics, and computer graphics to solve problems involving circular or cylindrical objects. They have advantages such as simplifying calculations, allowing for more intuitive representation, and being easier to work with
  • #1
joemama69
399
0

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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  • #2
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.
 
  • #3
but there is no x value in the function.
 
  • #4
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.
 

1. What are cylindrical coordinates?

Cylindrical coordinates are a three-dimensional coordinate system used to locate points in space using a combination of distance from a central axis, an angle from a fixed reference plane, and a height or depth from the reference plane. They are most commonly used in mathematics and engineering.

2. How do cylindrical coordinates differ from Cartesian coordinates?

In cylindrical coordinates, a point is specified by its distance from a central axis (r), an angle from a fixed reference plane (theta), and a height or depth from the reference plane (z). In Cartesian coordinates, a point is specified by its distance along three perpendicular axes (x, y, z). Cylindrical coordinates are often used when dealing with circular or cylindrical objects, while Cartesian coordinates are used for general purposes.

3. What is the conversion formula between cylindrical and Cartesian coordinates?

The conversion formula between cylindrical and Cartesian coordinates is: x = r*cos(theta), y = r*sin(theta), z = z. This means that to convert from cylindrical coordinates to Cartesian coordinates, you multiply the distance from the axis (r) by the cosine of the angle (theta) to get the x-coordinate, and multiply it by the sine of the angle to get the y-coordinate. The z-coordinate remains the same.

4. How are cylindrical coordinates used in real-world applications?

Cylindrical coordinates are commonly used in engineering, physics, and mathematics to solve problems involving circular or cylindrical objects. They are also used in computer graphics and game development to represent three-dimensional objects and their rotation. In real-world applications, cylindrical coordinates can be used to calculate the position, velocity, and acceleration of objects moving in circular or cylindrical paths.

5. What are the advantages of using cylindrical coordinates?

Cylindrical coordinates are useful for solving problems involving cylindrical or circular objects and can simplify calculations in these situations. They also allow for a more intuitive representation of certain systems, such as cylindrical tanks or pipes. In addition, cylindrical coordinates are often easier to visualize and work with in three-dimensional space compared to Cartesian coordinates.

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