To convert the equation x² + y² + z² = 49 into spherical coordinates, we need to recall the relationships between Cartesian and spherical coordinates. In spherical coordinates, the variables are defined as follows:
- x =
ρ sin(φ) cos(θ) - y =
ρ sin(φ) sin(θ) - z =
ρ cos(φ)
Here, ρ represents the radial distance from the origin, φ is the polar angle measured from the positive z-axis, and θ is the azimuthal angle in the x-y plane from the positive x-axis.
When we substitute these definitions into the left side of our equation, we get:
x² + y² + z² = (ρ sin(φ) cos(θ))² + (ρ sin(φ) sin(θ))² + (ρ cos(φ))²
Expanding this, we have:
ρ² sin²(φ) cos²(θ) + ρ² sin²(φ) sin²(θ) + ρ² cos²(φ)
This simplifies to:
ρ² (sin²(φ) (cos²(θ) + sin²(θ)) + cos²(φ))
Using the Pythagorean identity cos²(θ) + sin²(θ) = 1, the equation further simplifies to:
ρ² (sin²(φ) + cos²(φ)) = ρ²
Therefore, we can rewrite the original equation x² + y² + z² = 49 as:
ρ² = 49
This leads us to the final expression:
ρ = 7
In conclusion, the equation in spherical coordinates is simply ρ = 7, which describes a sphere of radius 7 centered at the origin.