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265Lesson04Problems - ρ = 2 ≤ θ ≤ π 2 and 0 ≤ φ...

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MATH 265: MULTIVARIABLE CALCULUS: LESSON 4 PROBLEMS Problems are worth 20 points each. (1) The point P has Cartesian coordinates ( 2 , 1 , 1). Find (a) the cylindrical coordi- nates of P , and (b) the spherical coordinates of P . (2) Sketch the set of points in space satisfying the cylindrical coordinate conditions 1 r 2, 0 θ π 2 , and 1 z 2. (3) Use cylindrical coordinates to describe the line through the point (1 , 1 , 0) and par- allel to the z -axis. (This is the reverse of problem 2 in the sense that you need to specify the conditions r , θ , and z need to satisfy.) (4) Sketch the set of points in space satisfying the spherical coordinate conditions
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Unformatted text preview: ρ = 2, ≤ θ ≤ π 2 , and 0 ≤ φ ≤ π 4 . (5) Use spherical coordinates to describe the region above the xy-plane between the spheres of radius 1 and 3 centered at the origin. (This is the reverse of problem 4.) (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) Determine the Cartesian equation of the surface with spherical coordinate equa-tion ρ = 2 cos θ sin φ-2 sin θ sin φ + 2 cos φ . It turns out this describes a sphere. What is the center and radius of this sphere? 1...
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