Cylindrical coordinates

This is a conversation where the person is discussing an equation and how the solutions manual determined the range for one of the variables. It is also mentioned that the person was looking at a solution to a different problem. In summary, the conversation is about how the equation x2 + y2 = 2y was solved and how the range of z was determined in the solutions manual. The solution to the equation involves finding the equation of a half-cylinder with restrictions on the given angle interval. The person is also mentioned to have been looking at a solution to a different problem.
  • #1
Miike012
1,009
0
From this equation

x2 + y2 = 2y

I was wondering how in the solutions manual it was decided that 0≤z≤1 ?

Edit:

Don't read... I was looking at a solution to a different problem
 

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  • #2
so the problem is solved?
 
  • #3
Miike012 said:
From this equation

x2 + y2 = 2y

I was wondering how in the solutions manual it was decided that 0≤z≤1 ?

Edit:

Don't read... I was looking at a solution to a different problem

Based on the thumbnail you just posted, we have the radii between 0 and 2. That gives the surfaces of the circle. If 0 ≤ z ≤ 1, then, we have the cylinder with the height between these intervals (The z range 1).

I assume that you want to find the equation of the half-cylinder.

Since r² = x² + y²...

0 ≤ r² = x² + y² ≤ 4

So we have x² + y² = 4 with origin as the center. That is the equation of the circular cylinder with radius 2. With restrictions of the given angle interval, the region of the cylinder occur in Quad. I and IV. This gives us the half-cylinder.

Hence, we have x² + y² = 4 or r = 2 where -π/2 ≤ θ ≤ π/2.

Let me know if this is what you are referring to. ;)
 
  • #4
The OP was looking at the solution to a different problem.

Thread locked.
 

1. What are cylindrical coordinates?

Cylindrical coordinates are a type of coordinate system used in mathematics and physics to represent points in three-dimensional space. They are based on the distance from a central axis, an angle from a reference direction, and the height above a reference plane.

2. How do cylindrical coordinates differ from Cartesian coordinates?

In cylindrical coordinates, a point is represented by its distance from a central axis, an angle from a reference direction, and a height above a reference plane. In Cartesian coordinates, a point is represented by its distance from the x, y, and z axes. Cylindrical coordinates are better suited for describing objects with cylindrical symmetry, while Cartesian coordinates are more commonly used for general three-dimensional space.

3. What are the advantages of using cylindrical coordinates?

Cylindrical coordinates are useful for simplifying calculations in problems involving cylindrical symmetry, such as in electromagnetism or fluid dynamics. They also allow for a more intuitive understanding of certain physical systems, such as rotating objects or cylindrical containers.

4. How do you convert between cylindrical and Cartesian coordinates?

To convert from cylindrical to Cartesian coordinates, use the following formulas: x = r * cos(θ), y = r * sin(θ), z = z. To convert from Cartesian to cylindrical coordinates, use: r = √(x^2 + y^2), θ = arctan(y/x), z = z. It is also important to note that the ranges of values for θ and z may need to be adjusted depending on the problem.

5. What are some real-world applications of cylindrical coordinates?

Cylindrical coordinates are commonly used in physics and engineering to describe the motion and properties of rotating objects, such as wheels or turbines. They are also used in calculating the flow of fluids in pipes or channels, and in analyzing the electromagnetic fields of cylindrical antennas or solenoids.

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