Cylindrical and spherical coordinates are alternative ways to describe points in 3D space when Cartesian coordinates make a problem harder than it needs to be. Cylindrical coordinates are useful for objects with circular symmetry around an axis, such as pipes, tanks, and rotating solids. Spherical coordinates are useful for objects with symmetry around a point, such as balls, planets, and fields spreading outward from a source.
Choosing the right coordinate system can turn a complicated triple integral into a much simpler one.
Key Facts
- Cylindrical coordinates: x = r cos θ, y = r sin θ, z = z.
- Cylindrical volume element: dV = r dz dr dθ, with r ≥ 0.
- Spherical coordinates: x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.
- Spherical volume element: dV = ρ^2 sin φ dρ dφ dθ, with ρ ≥ 0 and 0 ≤ φ ≤ π.
- Distance from the z-axis is r = sqrt(x^2 + y^2), while distance from the origin is ρ = sqrt(x^2 + y^2 + z^2).
- Use cylindrical coordinates for symmetry around an axis and spherical coordinates for symmetry around a center point.
Vocabulary
- Cylindrical coordinates
- A 3D coordinate system that locates a point using distance r from the z-axis, angle θ in the xy-plane, and height z.
- Spherical coordinates
- A 3D coordinate system that locates a point using distance ρ from the origin, polar angle φ from the positive z-axis, and azimuthal angle θ in the xy-plane.
- Jacobian
- A scaling factor that adjusts area or volume when changing from one coordinate system to another.
- Volume element
- The small piece of volume used inside a triple integral, such as dV = r dz dr dθ or dV = ρ^2 sin φ dρ dφ dθ.
- Azimuthal angle
- The angle θ measured in the xy-plane from the positive x-axis toward the positive y-axis.
Common Mistakes to Avoid
- Forgetting the factor r in cylindrical integrals is wrong because dV is not dz dr dθ, but r dz dr dθ due to the stretching of circular shells.
- Forgetting the factor ρ^2 sin φ in spherical integrals is wrong because spherical volume pieces get larger as distance from the origin and angle from the pole increase.
- Mixing up φ and θ is wrong because θ rotates around the z-axis while φ is measured down from the positive z-axis in the standard calculus convention.
- Using spherical coordinates for every round-looking problem is wrong because cylindrical coordinates are often simpler for tubes, cylinders, and regions with constant height or symmetry around an axis.
Practice Questions
- 1 Convert the point with cylindrical coordinates (r, θ, z) = (4, π/3, 5) into Cartesian coordinates.
- 2 Find the volume of a cylinder of radius 3 and height 8 using cylindrical coordinates by evaluating ∫ from 0 to 2π ∫ from 0 to 3 ∫ from 0 to 8 r dz dr dθ.
- 3 A solid ball is centered at the origin, but a vertical cylinder is centered on the z-axis. Explain which coordinate system is better for setting up the volume integral for each solid and why.