Spherical Coordinates for Sphere Between Z-Planes

  • Thread starter fball558
  • Start date
  • Tags
    Spherical
In summary, the problem is to find a parametric representation for the part of a sphere that lies between two planes. The solution involves using trigonometric functions to find the bounds for the angle ϕ, with the lower bound being 2pi/3 and the upper bound being pi. This allows for a full representation of the sphere within the given parameters.
  • #1
fball558
147
0

Homework Statement



Find a parametric representation for the part of the sphere x^2 + y^2 + z^2 = 64 that lies between the planes z = -4 and z = 4.

i have found
x = 8sin(ϕ)cos(θ)
y = 8sin(ϕ)sin(θ)
z= 8cos(ϕ)

0 ≤ θ ≤ 2π (n looking thing is pi)

now i need to find the bounds of ϕ

i know under perfect conditions they are 0 to π
but these are not perfect conditions and I am not sure how to find them exactly?
 
Physics news on Phys.org
  • #2
nevermind i got it.
what you do is take your z component and set it equal to you plane so
8cos(ϕ) = 4 you get cos(ϕ)=.5 so you get ϕ= pi/3
then by symmetry you know your lower bound will be pi - (pi/3) which give you 2pi/3
these are the correct bounds :)
 

FAQ: Spherical Coordinates for Sphere Between Z-Planes

What are spherical coordinates?

Spherical coordinates are a system of coordinates used to locate points in three-dimensional space. They are often used in scientific and mathematical applications, such as in astronomy and physics.

How are spherical coordinates represented?

Spherical coordinates are typically represented by three values: radius (r), inclination (θ), and azimuth (φ). These values describe a point's distance from the origin, its angle from the polar axis, and its angle from the reference plane, respectively.

What is the difference between spherical and Cartesian coordinates?

Spherical coordinates use a different system of measurement than Cartesian coordinates, which use the x, y, and z axes. Spherical coordinates use a radial distance, an angle from the polar axis, and an angle from the reference plane to describe a point's location.

What are some practical applications of spherical coordinates?

Spherical coordinates are commonly used in fields such as astronomy, navigation, and engineering. They are also useful for visualizing and analyzing three-dimensional data.

How do you convert between spherical and Cartesian coordinates?

To convert from spherical to Cartesian coordinates, you can use the following formulas:
x = r * sin(θ) * cos(φ)
y = r * sin(θ) * sin(φ)
z = r * cos(θ)
To convert from Cartesian to spherical coordinates, you can use these formulas:
r = √(x² + y² + z²)
θ = arccos(z/r)
φ = arctan(y/x)

Similar threads

Mass of a sphere with a non-uniform density.
Replies
6
Views
4K
Element of surface area in spherical coordinates
Replies
7
Views
4K
Surface area of a shifted sphere in spherical coordinates
Replies
3
Views
3K
Triple Integral To Find Volume Between Cylinder And Sphere
Replies
2
Views
400
Volume integral of x^2 + (y-2)^2 +z^2 = 4 where x , y , z > 0
Replies
21
Views
2K
Multivariable calculus, Integral using spherical coordinates
Replies
9
Views
3K
Find the volume of the region D using spherical coordinates
Replies
6
Views
2K
Find the Radius and Center of a Sphere, Quadric Surfaces
Replies
2
Views
2K
How to calculate a sink using spherical coordinates
Replies
7
Views
1K
Center of mass of spherical shell inside of cone (Apostol Problem).
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top