Math 233 - December 1, 2009 Spherical coordinates 1. Find the - - PowerPoint PPT Presentation

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Math 233 - December 1, 2009 Spherical coordinates 1. Find the - - PowerPoint PPT Presentation

Dec 10, 2022 108 likes •190 views

Math 233 - December 1, 2009 Spherical coordinates 1. Find the determinant: sin cos cos cos sin sin sin sin cos sin sin cos = cos

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SLIDE 1

Math 233 - December 1, 2009

◮ Spherical coordinates

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SLIDE 2
  • 1. Find the determinant:
  • sin φ cos θ

ρ cos φ cos θ −ρ sin φ sin θ sin φ sin θ ρ cos φ sin θ ρ sin φ cos θ cos φ −ρ sin φ

  • =
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SLIDE 3
  • 1. Find the determinant:
  • sin φ cos θ

ρ cos φ cos θ −ρ sin φ sin θ sin φ sin θ ρ cos φ sin θ ρ sin φ cos θ cos φ −ρ sin φ

  • = ρ2 sin φ
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SLIDE 4

Lecture Problems

  • 2. Convert the spherical equation to a Cartesian equation: ρ = sec φ
  • 3. Convert the Cartesian equation to a spherical equation:

x2 + y2 + 4z2 = 10

  • 4. Convert the Cartesian equation to a spherical equation:

x2 + y2 − 2z2 = 0

  • 5. Convert the Cartesian equation to a spherical equation:

x + y + z = 1

  • 6. Convert the spherical equation to a Cartesian equation: ρ sin φ = 1
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SLIDE 5

Lecture Problems

  • 2. Convert the spherical equation to a Cartesian equation: ρ = sec φ

Solution: z = 1

  • 3. Convert the Cartesian equation to a spherical equation:

x2 + y2 + 4z2 = 10 Solution: ρ = (3/2) sec φ

  • 4. Convert the Cartesian equation to a spherical equation:

x2 + y2 − 2z2 = 0 Solution: tan φ = 1/ √ 2

  • 5. Convert the Cartesian equation to a spherical equation:

x + y + z = 1 Solution: ρ sin φ cos θ + ρ sin φ sin θ + ρ cos φ = 1

  • 6. Convert the spherical equation to a Cartesian equation: ρ sin φ = 1

Solution: x2 + y2 = 1

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SLIDE 6
  • 7. Find the volume of the ice cream cone–the solid below the sphere

x2 + y2 + z2 = 1 and above the cone z2 = x2 + y2. V =

  • 8. Find the mass of the solid inside the sphere ρ = 3 and outside the

sphere ρ = 2 with density equal to the distance from the origin. m = 9. 3

−3

√

9−x2 − √ 9−x2

√

9−x2−z2 − √ 9−x2−z2(x2 + y2 + z2)3/2 dy dz dx

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SLIDE 7
  • 7. Find the volume of the ice cream cone–the solid below the sphere

x2 + y2 + z2 = 1 and above the cone z2 = x2 + y2. V = 2π π/4 1 ρ2 sin φ dρ dφ dθ = π(2 − √ 2) 3

  • 8. Find the mass of the solid inside the sphere ρ = 3 and outside the

sphere ρ = 2 with density equal to the distance from the origin. m = 2π π 3

2

ρρ2 sin φ dρ dφ dθ = 65π 9. 3

−3

√

9−x2 − √ 9−x2

√

9−x2−z2 − √ 9−x2−z2(x2 + y2 + z2)3/2 dy dz dx

= 2π

2

π 3 ρ3/2ρ2 sin φ dρ dφ dθ = 486π