Question

\[ x=r\sin\theta\cos\phi,\quad y=r\sin\theta\sin\phi,\quad z=r\cos\theta \]

Then

\[ x^2+y^2+z^2 \]

is independent of

(a) \(\theta,\phi\)
(b) \(r,\theta\)
(c) \(r,\phi\)
(d) \(r\)

Solution

\[ x^2+y^2+z^2 \]

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

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

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

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

\[ =r^2 \]

Hence, it is independent of \[ \theta \text{ and } \phi \]

Answer

\[ \boxed{\theta,\phi} \]

Correct Option: (a)