How to desribe a region in cylindrical coordinates

Pirito

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Jan 14, 2020
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2
I have to describe this region in cylindrical coordinates

Ω = {(x,y,z) ∈ R3 ; x2+y2+z2≤1 ; x2+y2≥1−z ; x>0 ; y>0 ; z>0}

With x>0 and y>0 I get θ∈(0,π/2).

My problem comes with the two first restrictions: I know x2+y2=r2, so I have:
  • r2≤1-z2
  • r2≥1-z

And from now on I don't know how to continue to resolve r.
Any help would be much appreciated.
 

Dr.Peterson

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Nov 12, 2017
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8,887
Why do you need to do more? Those are inequalities in cylindrical coordinates, so haven't you done what was asked? Or, if you wish, you could take square roots.

Just list four (technically, five) inequalities as your answer.
 

Pirito

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Jan 14, 2020
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So, the answer would be like Ω = {(r,θ,z) / 1-z≤r2≤1-z2, θ∈(0,π/2), z>0}, right?
 

Dr.Peterson

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Nov 12, 2017
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8,887
Yes, that's one way to say it.
 
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