MAST20029 · Engineering Mathematics
Polar, Cylindrical and Spherical Coordinates
Many integrals that are ugly in x, y, z become simple in a coordinate system matched to the region's symmetry. This chapter covers the general change-of-variables formula with its Jacobian and the three workhorses: polar (J=r) for disks and annuli, cylindrical (J=r) for cylinders, and spherical (J=r²sin φ, in the course's x=r cos θ sin φ convention) for balls. The examiner tests whether you pick the right system, set correct limits, and — above all — never forget the Jacobian.
What this chapter covers
- 011. The change-of-variables formula ∬_R f dx dy = ∬ f(x(u,v),y(u,v)) |J| du dv
- 022. The Jacobian J = ∂(x,y)/∂(u,v) and why |J| corrects the area element
- 033. Polar coordinates x=r cos θ, y=r sin θ, J=r, dA = r dr dθ
- 044. Cylindrical coordinates x=r cos θ, y=r sin θ, z=z, J=r (triple integrals)
- 055. Spherical coordinates x=r cos θ sin φ, y=r sin θ sin φ, z=r cos φ, J=r² sin φ
- 066. The course's angle convention: φ measured from the +z axis, θ the azimuth
- 077. Matching the system to the region's symmetry (disk → polar, ball → spherical)
- 088. Setting θ, φ, r limits from the region BEFORE integrating
Mass of a quarter-annulus in polar coordinates
- 1 markThe region is a quarter-annulus, so switch to polar: x² + y² = r², dA = r dr dθ. The bounds become 1 ≤ r ≤ 3 (from 1 ≤ r² ≤ 9) and 0 ≤ θ ≤ π/2 (the first quadrant).
- 1 markRewrite the density and combine with the Jacobian: ρ = 1/r², so the integrand ρ · r = (1/r²) · r = 1/r. This is the key step — the Jacobian r is exactly what tames the 1/r² density.
- 1 markSeparate the integral (the integrand factors): ∫₀^(π/2) dθ · ∫₁³ (1/r) dr = (π/2) · [ln r]₁³ = (π/2)(ln 3 − ln 1) = (π/2) ln 3.
- 1 markState the mass: (π/2) ln 3.
Key terms
- Change of variables
- Replacing x, y (or x, y, z) by new coordinates u, v matched to the region; the integral picks up the Jacobian factor |J| to correct the distorted area/volume element.
- Jacobian J
- The determinant ∂(x,y)/∂(u,v) (or its 3D analogue). |J| scales du dv into the true area element: polar/cylindrical give J=r; spherical gives J=r² sin φ.
- Polar coordinates
- x = r cos θ, y = r sin θ, with J = r so dA = r dr dθ. Ideal for disks, annuli and any region with a circular boundary or an x²+y² integrand.
- Cylindrical coordinates
- Polar in the xy-plane plus z unchanged: x = r cos θ, y = r sin θ, z = z, J = r. The natural system for cylinders and solids with an axis of symmetry.
- Spherical coordinates
- In this course's convention x = r cos θ sin φ, y = r sin θ sin φ, z = r cos φ with J = r² sin φ; φ is measured from the +z axis and θ is the azimuthal angle. Used for balls and spheres.
- Symmetry matching
- The habit of choosing the coordinate system from the region's shape — circular → polar, cylindrical → cylindrical, spherical → spherical — so that the limits become constants and the integrand simplifies.
Polar, Cylindrical and Spherical Coordinates FAQ
How do I know which coordinate system to use?
Let the region's symmetry decide. A circular or annular boundary, or an integrand built from x²+y², points to polar; a cylinder or an axis of symmetry points to cylindrical; a ball, sphere or an x²+y²+z² integrand points to spherical. The right system turns curved limits into constant ones and usually simplifies the integrand.
What exactly is the Jacobian and why must I include it?
The Jacobian |J| corrects the size of the area or volume element when you change coordinates — a small box in the new variables maps to a region of a different size in x, y, z. Polar and cylindrical give |J| = r; spherical gives |J| = r² sin φ. Forgetting it is the single biggest source of lost marks in this chapter and can even turn a convergent integral into a divergent one.
What is the course's spherical convention?
This subject uses x = r cos θ sin φ, y = r sin θ sin φ, z = r cos φ, where φ is the angle measured down from the +z axis and θ is the azimuthal angle in the xy-plane, with Jacobian r² sin φ. Different texts swap the roles of φ and θ, so always use the convention on the provided formula sheet to keep your limits consistent.
How are coordinate changes examined?
As the core of Q1–Q3 in the vector-calculus block: you are given a region and an integral, and you must pick the system, set the limits, include the Jacobian, and evaluate. Marks reward the right choice of system, correct limits read from the geometry, and the Jacobian — the integration itself is usually short once the setup is right.
Exam move
Internalise three pairings — disk/annulus → polar (J=r), cylinder → cylindrical (J=r), ball → spherical (J=r² sin φ) — and make including the Jacobian an unbreakable habit; write dA = r dr dθ (or dV = r² sin φ dr dφ dθ) as the very first line so you cannot forget it. Read the limits from the region: angles from the quadrant or sector, radius from the boundary curves, and only then integrate. Memorise the course's spherical convention (φ from the +z axis) exactly as the formula sheet states it, since other books differ. Drill setup-only on several regions until choosing the system and writing the limits is automatic — that is where the marks are.
