Describing a region using spherical coordinates

In summary, the solid E in the first octant that lies above the half-cone z=√(x2+y2) but inside x2+y2+z2=1 can be described using spherical coordinates as z=√(x2+y2) ≤ ρ ≤ 1, with θ going from [0, π/2] and φ being equal to θ. The lower bound for ρ is determined by setting tan φ < π/4, which is found by picking a point in the region and varying ρ until hitting the boundaries of the region.
  • #1
Kaleem
21
0

Homework Statement


Describe using spherical coordinates the solid E in the first octant that lies above the half-cone z=√(x2+y2) but inside x2+y2+z2=1. Your final answer must be written in set-builder notation.

Homework Equations


ρ = x2+y2+z2
x = ρsinφcosθ
y = ρsinφsinθ
z = ρcosφ

The Attempt at a Solution


Since we are in the first octant, θ will go from [0,π/2].
However the problem comes with describing ρ and φ,
Since we are in the first octant I believe that φ will be the same as θ, however for ρ,
I substituted in the relevant equations into both equations that were given.

ρ= ±1 <-- Unit sphere
ρ=√(2ρ2sin2φ)/cosφ <-- Half cone

Would I be able to use the positive bound of the unit sphere as the upper limit and the bound gotten from the cone as the lower limit?
 
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  • #2
Kaleem said:
Would I be able to use the positive bound of the unit sphere as the upper limit and the bound gotten from the cone as the lower limit?
Yes.
Kaleem said:
x = ρsinφcosθ
y = ρsinφcosθ
One of those should be sin θ.
Kaleem said:
ρ=√(2ρ2sin2φ)/cosφ
Where does the 2 come from? Can you not simplify this?
 
  • #3
haruspex said:
Yes.

One of those should be sin θ.

Where does the 2 come from? Can you not simplify this?
You're right, I made a mistake when i factored it out, if i simplify I get 1 = tanφ, which gives me φ= π/4. Which confuses me, would the lower bound for ρ be π/4 or would that just be the upper bound for φ?
 
  • #4
Kaleem said:
would the lower bound for ρ be π/4 or would that just be the upper bound for φ?
how does tan φ < π/4 turn into a bound on ρ?
Pick a point in the region. Vary ρ up and down until you hit the boundaries of the region. Which boundaries do you hit?
 
  • #5
haruspex said:
how does tan φ < π/4 turn into a bound on ρ?
Pick a point in the region. Vary ρ up and down until you hit the boundaries of the region. Which boundaries do you hit?
You get z=√(x2+y2) ≤ ρ ≤1
 
  • #6
Kaleem said:
You get z=√(x2+y2) ≤ ρ
Although that is true, it does not serve as a limit on ρ in polar coordinates. The limits should not mention the Cartesian coordinates.
If the point (ρ, θ,φ) is in the region, is the point (ρ/2, θ, φ) also in the region?
 

1. What are spherical coordinates?

Spherical coordinates are a system used to describe a point in three-dimensional space using three coordinates: radius, inclination, and azimuth. It is often used in physics, astronomy, and engineering to specify the position of an object in space.

2. How do spherical coordinates differ from Cartesian coordinates?

In spherical coordinates, the position of a point is described using a radial distance, an angle from the positive z-axis, and an angle from the positive x-axis. In Cartesian coordinates, a point is described using its distance from the x, y, and z axes. Spherical coordinates are better suited for describing positions in spherical or cylindrical shapes, while Cartesian coordinates are better for describing positions in rectangular shapes.

3. What are the advantages of using spherical coordinates?

Spherical coordinates have several advantages over other coordinate systems. They are particularly useful for describing positions on a sphere or a portion of a sphere, which is common in many scientific and engineering applications. They are also useful for visualizing and analyzing spherical or cylindrical objects. Additionally, spherical coordinates are often used in calculations involving vectors and rotations, making them essential in fields such as mechanics and physics.

4. How are spherical coordinates converted to Cartesian coordinates?

To convert spherical coordinates (r, θ, φ) to Cartesian coordinates (x, y, z), the following equations can be used: x = r sinθ cosφ, y = r sinθ sinφ, and z = r cosθ. These equations can be derived from the Pythagorean theorem and trigonometric functions.

5. Can spherical coordinates be used to describe a point on the Earth's surface?

Yes, spherical coordinates can be used to describe any point on the Earth's surface. The radius coordinate (r) would represent the distance from the center of the Earth, the inclination angle (θ) would represent the latitude, and the azimuth angle (φ) would represent the longitude. However, for practical purposes, other coordinate systems such as latitude and longitude are often used instead of spherical coordinates.

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