Consider the sphere x2 + y2 + z? = 4. In cylindrical coordinates this sphere can be expressed as r2 + z² = 4. Explain why. In spherical coordinates this sphere can be expressed as p = 2. Explain why.

Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
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Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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This is a calculus 3 problem. Please explain each step clearly, no cursive writing.

 

1. Consider the sphere x2 + y2 +z² = 4.
a) In cylindrical coordinates this sphere can be expressed as r² +z = 4. Explain why.
b) In spherical coordinates this sphere can be expressed as p
%3|
2. Explain why.
Transcribed Image Text:1. Consider the sphere x2 + y2 +z² = 4. a) In cylindrical coordinates this sphere can be expressed as r² +z = 4. Explain why. b) In spherical coordinates this sphere can be expressed as p %3| 2. Explain why.
Expert Solution
Step 1

Given Sphere is x2+y2+z2=4

Explanation of sphere in cylindrical and spherical coordinates

Step 2

Cylindrical Coordinates:

x=r cosθ , y=r sinθ and z=z

r0 and θ[0,2π)

Let us substitute in given sphere

(r cosθ)2+r sinθ2+z2=4

r2cos2θ+r2sin2θ+z2=4r2(cos2θ+sin2θ)+z2=4r2+z2=4      (cos2θ+sin2θ=1)

 

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