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.
Understanding Calculus: Cylindrical and Spherical Coordinates
A coordinate change does more than rename a point. It changes the shape of the tiny pieces used to build an integral. In a rectangular grid, a tiny volume is a box with nearly constant side lengths.
Around an axis, equal changes in angle make wider pieces when they are farther from the axis. This is why cylindrical integration includes a radial scale factor. Near the axis, an angular slice is narrow.
Farther out, the same slice covers more distance. Ignoring that change would count outer regions as if they had the same volume as inner regions.
Spherical coordinates need an extra scale adjustment because two directions are measured by angles. A small change in the direction around the vertical axis traces a circle whose size depends on distance from the origin. A small change in the vertical angle traces another curved path.
Together, these effects make a small spherical volume grow rapidly with distance. The factor involving the sine of the vertical angle has a physical meaning. Near the top and bottom of a sphere, circular paths are very small.
Near the middle, they are largest. This prevents narrow regions near the poles from being counted too heavily.
The main skill is usually setting correct bounds. Start by describing the solid in ordinary words before writing any integral. State how far it extends from its central axis or center, which directions it covers, and where it begins and ends vertically.
A full rotation uses all horizontal directions, while a half rotation covers only one side. Sketching a top view is especially useful for cylindrical problems. Sketching a vertical cross section through the origin helps with spherical problems.
Curved boundaries often become simple limits, but planes and cones require careful translation. A cone through the origin can often be described by a fixed vertical angle. A sphere centered at the origin can often be described by a fixed maximum distance.
Students meet these ideas whenever a quantity is spread through a three dimensional region. Density can vary inside a tank, temperature can vary inside a metal ball, and electric or gravitational effects can depend only on distance from a source. The integral adds contributions from many tiny volumes to find mass, charge, energy, or average value.
Pay close attention to angle conventions. In most calculus courses, the horizontal angle is measured in the plane and the vertical angle is measured down from the positive vertical axis. Some science courses use a different vertical angle convention.
Check the diagram and definitions before substituting values. Also check units. Every scale factor has a length hidden inside it, so the final integral must have the units expected for volume or for the physical quantity being calculated.
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.