Question

In Exercises \(55-60\), convert the rectangular equation to an equation in (a) cylindrical coordinates and (b) spherical coordinates $$ x^{2}+y^{2}=16 $$

Short answer

The equivalent equation in cylindrical coordinates is \(r^{2}\) = 16 and in spherical coordinates is \(ρ^{2}\) = 16.

Step-by-step solution

Convert to Cylindrical Coordinates

Cylindrical coordinates are composed of (r, θ, z), where \(r^{2}\) = \(x^{2}\) + \(y^{2}\). Plugging this into our original equation, we get: \(r^{2}\) = 16.

Convert to Spherical Coordinates

Spherical coordinates are composed of (ρ, θ, φ), where \(ρ^{2}sin^{2}φ\) = \(x^{2}\) + \(y^{2}\). Substituting this into the original equation, we get: \(ρ^{2}sin^{2}φ\) = 16. Since there is no z-component, φ = π/2, we simply get \(ρ^{2}\) = 16

Final Cylindrical and Spherical Equations

So, in cylindrical coordinates, the equation becomes: \(r^{2}\) = 16. In spherical coordinates, the equation becomes \(ρ^{2}\) = 16