{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

1211s2

1211s2 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

This preview shows pages 1–2. Sign up to view the full content.

THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 1st Semester: Solution to Assignment 2 1. Using Cartesian coordinates, we have x = ρ sin ϕ cos θ, y = ρ sin ϕ sin θ, z = ρ cos ϕ and the given equation ρ sin ϕ sin θ = 2 is translated to y = 2. In Cylindrical coordinates, r = ρ sin ϕ, θ = θ, z = ρ cos ϕ and the equation is translated to r sin θ = 2. The surface is a vertical plane. 2. Since p x 2 + y 2 + z 2 = ρ , from 0 ρ 2 we obtain 0 p x 2 + y 2 + z 2 2. The solid is a portion of the sphere of radius 2. Also we have 0 ρ π 4 , hence, the solid is an ice-cream- cone-like solid in R 3 3. Assume that the centre of the sphere (that is, the completion of hemisphere) is positioned at the origin. The solid is, when using Spherical coordinates, the set: { ( ρ, ϕ, θ ) | 0 ρ 5 , π/ 2 ρ π, 0 θ < 2 π } When using Cylindrical coordinates, the solid is the set: { ( r, θ, z ) | z 2 + r 2 5 , z 0 , 0 θ < 2 π } 4. (a) Evaluating the limit along the line x = 0, we obtain lim ( x,y ) (0 , 0) , x =0 2 xy x 2 + y 2 = lim y 0 0 y 2 = 0 . Evaluating the lime along the line x = y , we obtain lim ( x,y ) (0 , 0) , x = y 2 xy x 2 + y 2 = lim x 0 2 x 2 x 2 + x 2 = 1 . Because these two limits are not equal, we conclude that lim ( x,y ) (0 , 0) , x =0 2 xy x 2 + y 2 does not exist. (Remark : You may use polar form to solve this problem.) (b) Note that when ( x, y, z ) (0 , 0 , 0), we have y 0, y 0, and z 0 respectively. Note also 0 y 3 x 2 + y 2 + z 4 = | y | y 2 x 2 + y 2 + z 4 ≤ | y | . Here the last inequality follows because (0 ) y 2 x 2 + y 2 + z 4 1. Thus when ( x, y, z ) (0 , 0 , 0), we have x 0, and hence y 3 x 2 + y 2 + z 4 0 also, i.e., lim ( x,y,z ) (0 , 0 , 0) y 3 x 2 + y 2 + z 4 = 0.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 4

1211s2 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online