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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 isฯ=cscฯ• .

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=ฯsinฯ•cosฮธ

y=ฯsinฯ•sinฮธ

z=ฯcosฯ•

Using values of x,yin equation of cylinder,

(ฯsinฯ•cosฮธ)2+(ฯsinฯ•sinฮธ)2=1

ฯ2sin2ฯ•cos2ฮธ+ฯ2sin2ฯ•sin2ฮธ=1

ฯ2sin2ฯ•cos2ฮธ+sin2ฮธ=1

ฯ2sin2ฯ•=1

Therefore, we get

ฯ2=1sin2ฯ•

ฯ2=csc2ฯ•

Hence

ฯ=cscฯ•

Hence, proved.

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