How to Write the Equations in Spherical Coordinates: a x² + y² + z² = 81 and b x² + y² + z² = 1?

To convert the given equations into spherical coordinates, we start by recalling the relationship between Cartesian coordinates (x, y, z) and spherical coordinates (ρ, θ, φ). In spherical coordinates:

• x = ρ sin φ cos θ
• y = ρ sin φ sin θ
• z = ρ cos φ

Given the equations:

1. a) x² + y² + z² = 81
2. b) x² + y² + z² = 1

We know that in spherical coordinates, the equation x² + y² + z² corresponds to ρ². Thus, we can rewrite the equations:

1. a) ρ² = 81
2. b) ρ² = 1

To find the value of ρ for each equation, we take the square root:

1. a) ρ = √81 = 9
2. b) ρ = √1 = 1

Therefore, the equations in spherical coordinates become:

• a) ρ = 9
• b) ρ = 1

In these forms, the equations represent spheres centered at the origin with radii of 9 and 1, respectively. No further information about angles θ and φ is specified, as these are generally not constrained in such equations.