To convert the equation x² + y² + z² = 25 into spherical coordinates, we need to understand the relationship between Cartesian coordinates (x, y, z) and spherical coordinates (ρ, θ, φ).
In spherical coordinates, the relationships are defined as follows:
- x = ρ sin(φ) cos(θ)
- y = ρ sin(φ) sin(θ)
- z = ρ cos(φ)
Here, ρ is the radial distance from the origin, θ is the azimuthal angle, and φ is the polar angle.
Substituting the spherical expressions into the original equation:
x² + y² + z² = ρ² sin²(φ) cos²(θ) + ρ² sin²(φ) sin²(θ) + ρ² cos²(φ).
Combine the first two terms:
x² + y² = ρ² sin²(φ)(cos²(θ) + sin²(θ)).
Using the Pythagorean identity cos²(θ) + sin²(θ) = 1, it simplifies to:
x² + y² + z² = ρ² sin²(φ) + ρ² cos²(φ) = ρ² (sin²(φ) + cos²(φ)).
Again, using the identity sin²(φ) + cos²(φ) = 1, we get:
ρ² = 25.
Thus, we conclude:
ρ = 5.
Therefore, the equation x² + y² + z² = 25 in spherical coordinates can be expressed as:
ρ = 5