Q. If$\ x = r\sin{\theta\cos{\phi,\ y = r\sin{\theta\sin{\phi\ and\ \mathcal{z =}r\cos{\theta,\ }}}}}$ then the value of $x^{2} + y^{2} + \mathcal{z}^{2}$ is independent of
A) $\theta,\phi$
B) $r,\ \theta$
C) $r,\ \phi$
D) $r$
Explanation
(a)
$x^{2} + y^{2} + \mathcal{z}^{2} = r^{2}\sin^{2}\theta\cos^{2}{\phi + r^{2}\sin^{2}{\theta\sin^{2}{\phi + r^{2}\cos^{2}\theta}}}$
$= r^{2}\sin^{2}{\theta\left( \cos^{2}{\phi + \sin^{2}\phi} \right) + r^{2}\cos^{2}\theta}$
$= r^{2}\sin^{2}{\theta + r^{2}\cos^{2}\theta}$
$= r^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.