Questions and Answers
In which type of problems is the use of spherical coordinates particularly useful?
Problems with spherical symmetry
What is the typical order of integration in Cartesian coordinates?
dz dx dy or dx dy dz or dy dz dx
What is the conversion formula for x in spherical coordinates?
x = ρ sin(φ) cos(θ)
What is the importance of triple integration in real-world applications?
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What is the conversion formula for z in cylindrical coordinates?
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Study Notes
Triple Integration
Cylindrical Coordinates
- Also known as cylindrical polar coordinates
- Conversion from Cartesian coordinates:
- x = r cos(θ)
- y = r sin(θ)
- z = z
- Volume element: dV = r dr dθ dz
- Integration order: typically dr dθ dz
- Useful for problems with cylindrical symmetry
Spherical Coordinates
- Also known as spherical polar coordinates
- Conversion from Cartesian coordinates:
- x = ρ sin(φ) cos(θ)
- y = ρ sin(φ) sin(θ)
- z = ρ cos(φ)
- Volume element: dV = ρ^2 sin(φ) dρ dφ dθ
- Integration order: typically dρ dφ dθ
- Useful for problems with spherical symmetry
Cartesian Coordinates
- Rectangular coordinates (x, y, z)
- Volume element: dV = dx dy dz
- Integration order: typically dx dy dz or dy dz dx or dz dx dy
- Useful for problems with rectangular symmetry or no particular symmetry
General Triple Integration
- Iterated integrals: ∫∫∫ f(x, y, z) dx dy dz
- Fubini's theorem: allows iterated integrals to be computed in any order
- Importance: used in physics, engineering, and computer science to model and solve problems involving volume and surface integrals.
Triple Integration
Coordinate Systems
-
Cylindrical Coordinates
- Also known as cylindrical polar coordinates
- Conversion from Cartesian coordinates: x = r cos(θ), y = r sin(θ), z = z
- Volume element: dV = r dr dθ dz
- Integration order: typically dr dθ dz
- Useful for problems with cylindrical symmetry
-
Spherical Coordinates
- Also known as spherical polar coordinates
- Conversion from Cartesian coordinates: x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), z = ρ cos(φ)
- Volume element: dV = ρ^2 sin(φ) dρ dφ dθ
- Integration order: typically dρ dφ dθ
- Useful for problems with spherical symmetry
-
Cartesian Coordinates
- Rectangular coordinates (x, y, z)
- Volume element: dV = dx dy dz
- Integration order: typically dx dy dz or dy dz dx or dz dx dy
- Useful for problems with rectangular symmetry or no particular symmetry
Key Concepts
- Iterated Integrals: ∫∫∫ f(x, y, z) dx dy dz
- Fubini's Theorem: allows iterated integrals to be computed in any order
- Importance: used in physics, engineering, and computer science to model and solve problems involving volume and surface integrals
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