Physics I / Change of Variables and the Jacobian
Practice question · Match the pairs

Match each coordinate change to its volume/area factor (its Jacobian).

  • Polar (r,θ)(r, \theta)
  • Cylindrical (r,θ,z)(r, \theta, z)
  • Spherical (ρ,φ,θ)(\rho, \varphi, \theta)
  • Linear map u=ax+by, v=cx+dyu = ax + by,\ v = cx + dy
  • rr
  • rr (the zz direction adds no distortion)
  • 1/∣ad−bc∣1/|ad - bc| (from dx dydx\,dy to du dvdu\,dv)
  • ρ2sin⁡φ\rho^{2}\sin\varphi
Hints
  1. Each factor answers: how much does a tiny coordinate box distort into real area or volume?
  2. Spherical distorts twice — once for radius, once for latitude — hence two factors: ρ2\rho^{2} and sin⁡φ\sin\varphi.
Show the answer
  • Polar (r,θ)(r, \theta) → rr
  • Cylindrical (r,θ,z)(r, \theta, z) → rr (the zz direction adds no distortion)
  • Spherical (ρ,φ,θ)(\rho, \varphi, \theta) → ρ2sin⁡φ\rho^{2}\sin\varphi
  • Linear map u=ax+by, v=cx+dyu = ax + by,\ v = cx + dy → 1/∣ad−bc∣1/|ad - bc| (from dx dydx\,dy to du dvdu\,dv)
Why

Jacobians are exchange rates between coordinate area and true area. Cylindrical inherits polar’s rr untouched; spherical pays ρ2sin⁡φ\rho^{2}\sin\varphi (shells grow with radius, and longitude lines crowd at the poles); linear maps pay a constant determinant.

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