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# s2 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2011-12 Second Semester: Solution to Assignment 2 1. In cylindrical coordinates x = r cos θ, y = r sin θ, z = z, r 0 , 0 θ < 2 π, z R , the equation can be translated to z 2 = 2 r 2 . In spherical coordinates x = ρ sin ϕ cos θ, y = ρ sin ϕ sin θ, z = ρ cos ϕ, ρ 0 , 0 ϕ π, 0 θ < 2 π, the equation is be translated to ρ 2 cos 2 ϕ = 2 ρ 2 sin 2 ϕ which is equivalent to ρ = 0 or tan 2 ϕ = 1 / 2. Note that ρ = 0 can be included in tan 2 ϕ = 1 / 2. There the equation in spherical coordinates is tan ϕ = ± 1 2 ( ϕ = arctan 1 2 or π - arctan 1 2 ) . 2. The condition 0 ρ 2 / cos ϕ forces that cos ϕ > 0 and then 0 z = ρ cos ϕ 2. Note that under the condition cos ϕ > 0, we have 0 ϕ π/ 4 ⇐⇒ 0 tan ϕ 1 ⇐⇒ 0 p x 2 + y 2 z 1 Therefore we have p x 2 + y 2 z 2.

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s2 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS...

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