等效座標理論下的一個現像
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x = r sin(θ) cos(φ)
y = r sin(θ) sin(φ)
z = r cos(θ)

dx = dr sin(θ) cos(φ) + r cos(θ) cos(φ) dθ - r sin(θ) sin(φ) dφ
dy = dr sin(θ) sin(φ) + r cos(θ) sin(φ) dθ + r sin(θ) cos(φ) dφ
dz = dr cos(θ) - r sin(θ) dθ

x dx + y dy + z dz
= r sin(θ) cos(φ) (dr sin(θ) cos(φ) + r cos(θ) cos(φ) dθ - r sin(θ) sin(φ) dφ) +
r sin(θ) sin(φ) (dr sin(θ) sin(φ) + r cos(θ) sin(φ) dθ + r sin(θ) cos(φ) dφ) +
r cos(θ) (dr cos(θ) - r sin(θ) dθ)
= r dr (sin(θ))^2((cos(φ))^2 + (sin(φ))^2) + r^2 sin(θ) cos(θ) dθ((cos(φ))^2 + (sin(φ))^2) +
r^2 dφ (sin(θ))^2(-cos(φ) sin(φ) + cos(φ) sin(φ)) +
r dr (cos(θ))^2 - r^2 sin(θ) cos(θ) dθ
= r dr (sin(θ))^2 + r^2 sin(θ) cos(θ) dθ +
r dr (cos(θ))^2 - r^2 sin(θ) cos(θ) dθ
= r dr ((sin(θ))^2 + (cos(θ))^2)
= r dr

i.e. independent of θ and φ