1211a2 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 1st Semester: Assignment 2 Due date: 2010 October 5 (Tuesday), 5:00 pm. 1. Translate the equation ρ sin ϕ sin θ = 2 into equations in Cartesian and in cylindrical coor- dinates, respectively. Provide an appropriate sketch of the set of points which satisfy the equation(s). 2. Sketch the solid whose spherical coordinates ( ρ,ϕ,θ ) satisfy 0 ϕ π/ 4, 0 ρ 2. 3. Consider the solid hemisphere of radius 5 pictured in Figure 1.124 on Page 78. (a) Describe this solid, using spherical coordinates. (b) Describe this solid, using cylindrical coordinates. 4. Evaluate the limits in the following,or explain why the limit fails to exist. (a) lim ( x,y ) (0 , 0) 2 xy x 2 + y 2 (b) lim ( x,y,z ) (0 , 0 , 0) y 3 - 1000 xy 2 + z 5 x 2 + y 2 + z 4 5. Examine the behavior of f ( x,y ) = x 4 y 4 / ( x 2 + y 4 ) 3 as ( x,y ) approaches (0 , 0) along various straight lines. From your observations, what might you conjecture lim
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1211a2 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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