1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Describing a region using spherical coordinates

  1. Oct 12, 2016 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations
    ρ = x2+y2+z2
    x = ρsinφcosθ
    y = ρsinφsinθ
    z = ρcosφ

    3. 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?
     
    Last edited: Oct 12, 2016
  2. jcsd
  3. Oct 12, 2016 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Yes.
    One of those should be sin θ.
    Where does the 2 come from? Can you not simplify this?
     
  4. Oct 12, 2016 #3
    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?
     
  5. Oct 12, 2016 #4

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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?
     
  6. Oct 12, 2016 #5
    You get z=√(x2+y2) ≤ ρ ≤1
     
  7. Oct 12, 2016 #6

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted