( r , θ , ϕ ) ⇒ ( ρ , ϕ , z ) (r, \theta, \phi) \Rightarrow (\rho, \phi, z)

Algebra Level 3

Express ρ \rho in terms of spherical coordinates.

D e t a i l s a n d A s s u m p t i o n s Details \ and \ Assumptions :

  1. In this picture, the red one is the ϕ \phi -coordinate.

  2. The x − y − z − a x e s x-y-z-axes are oriented such that you are facing the first octant.

r cos ⁡ θ r\cos\theta r cos ⁡ ϕ r\cos\phi r sin ⁡ θ r\sin\theta r sin ⁡ ϕ r\sin\phi

JohnDonnie Celestre by JohnDonnie Celestre , 25, Philippines

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1 solution

Since x = r sin ⁡ θ cos ⁡ ϕ . . . e q . 1 x=r\sin\theta\cos\phi ... \ eq.1 y = r sin ⁡ θ sin ⁡ ϕ . . . e q . 2 y=r\sin\theta\sin\phi ...\ eq.2 and ρ 2 = x 2 + y 2 . . . e q . 3 \rho^2=x^2+y^2 ...\ eq.3

Square e q . 1 \ eq.1 and e q . 2 \ eq.2 and substitute to e q . 3 \ eq.3 That yields ρ = r sin ⁡ θ \boxed{\rho=r\sin\theta}

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