Question

If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then x² + y² + z² is independent of

Options

• θ, ϕ

• r, θ

• r, ϕ

• r

Answer 1

θ, ϕ
We have:
x = r sin θ cos ϕ  ,  y = r sin θ sin ϕ and z = r cos θ,
∴ x² + y² + z²

\[= \left( r \sin\theta \cos\phi \right)^2 + \left( r \sin\theta \sin\phi \right)^2 + \left( r \cos\theta \right)^2 \]

\[ = r^2 \sin^2 \theta \cos^2 \phi + r^2 \sin^2 \theta \sin^2 \phi + r^2 \cos^2 \theta \]

\[ = r^2 \sin^2 \theta \left( \cos^2 \phi + \sin^2 \phi \right) + r^2 \cos^2 \theta \]

\[ = r^2 \sin^2 \theta \times 1 + r^2 \cos^2 \theta\]

\[ = r^2 \sin^2 \theta + r^2 \cos^2 \theta\]

\[ = r^2 \left( \sin^2 \theta + \cos^2 \theta \right)\]

\[ = r^2 \times 1\]

\[ = r^2 \]

\[\text{ Thus, }x^2 + y^2 + z^2\text{ is independent of }\theta\text{ and }\phi .\]