Question

Determine the scale factors for spherical coordinates. Also find the arc and volume elements.

Answer 1

For spherical coordinates

x = r sinθ cosφ

y = r sinθ sinφ

z = r cosθ

Hence, dx = dr sinθcosφ + dθr cosθcosφ – dφ rsinθ sinφ ...(1)

dy = dr sin θ sin φ + d θ r cos θ sin φ + d φ r sin θ cos φ ...(2)

and dz = dr cos θ – d θ r sin θ ...(3)

Squaring and adding (1), (2) and (3), we have

dx² + dy² + dz² = ds² = dr² + r²dθ² + r²sin²θ d φ²

Hence, h₁  = 1, ⇒ h₁ = 1, h₂ = r, h₃ = r sin θ