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From Example 1, recall that x2+y2=1is the equation of the cylinder with radius 1, whose axis of symmetry is the z-axis. Show that the equation of this cylinder in spherical coordinates isρ=cscϕ .

Short Answer

Expert verified

It is solved by substituting the value ofx,yin terms of spherical coordinates.

Step by step solution

01

Given Information

The equation of cylinder is x2+y2=1with radius 1unit and axis of symmetry aszaxis.

02

Simplification

We know the relation:

x=ρsinϕcosθ

y=ρsinϕsinθ

z=ρcosϕ

Using values of x,yin equation of cylinder,

(ρsinϕcosθ)2+(ρsinϕsinθ)2=1

ρ2sin2ϕcos2θ+ρ2sin2ϕsin2θ=1

ρ2sin2ϕcos2θ+sin2θ=1

ρ2sin2ϕ=1

Therefore, we get

ρ2=1sin2ϕ

ρ2=csc2ϕ

Hence

ρ=cscϕ

Hence, proved.

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