Cylindrical => Spherical

ρ = √r^2 + z^2 θ = θ tanϕ =r/z

Rectangular => Cylindrical

r = √x^2 + y^2 tanθ = y/x z = z

Rectangular => Spherical

ρ = √x^2 + y^2 + z^2 tanθ = y/x cosϕ = z/√x^2 + y^2 + z^2

Spherical => Cylindrical

r = ρsinϕ θ = θ z = ρcosϕ

Spherical => Rectangular

x = ρsinϕcosθ y = ρsinϕsinθ z = ρcosϕ

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Calculus Coordinates

Spherical and Cylindrical Coordinates

Question	Answer
Cylindrical => Rectangular	x = rcosθ y = rsinθ z = z
Cylindrical => Spherical	ρ = √r^2 + z^2 θ = θ tanϕ =r/z
Rectangular => Cylindrical	r = √x^2 + y^2 tanθ = y/x z = z
Rectangular => Spherical	ρ = √x^2 + y^2 + z^2 tanθ = y/x cosϕ = z/√x^2 + y^2 + z^2
Spherical => Cylindrical	r = ρsinϕ θ = θ z = ρcosϕ
Spherical => Rectangular	x = ρsinϕcosθ y = ρsinϕsinθ z = ρcosϕ