Question

If $x=r\sin \theta \cos \varphi ,y=r\sin \theta \sin \theta \sin \varphi$ and $z=r\cos \theta$, then $x^2+y^2+z^2$ is independent of
(a) $\theta ,\varphi$
(b) $r,\theta$
(c) $r,\varphi$
(d) $r$

Answer 1

The correct option is (a) $(\theta ,\varphi )$
We have:
$x=r\sin \theta \cos \varphi$ , $y=r\sin \theta$ and $z=r\cos \theta$

Consider $x^2+y^2+z^2$
$=(r\sin \theta \cos \varphi {)}^2+{r\sin \theta }^2+{r\cos \theta }^2$
$=r^2{\sin }^2\theta {\cos }^2\varphi +r^2{\sin }^2\theta {\sin }^2\varphi +r^2{\cos }^2\theta$ = r^{2} \sin^{2}\theta(\cos^{2}\phi+\sin^{2}\phi)+r^2\cos^{2}\theta=r^{2}\sin^{2}\theta(1)+r^2\cos^{2}\theta $[Since,$(\cos^{2}\phi+\sin^{2}\phi)=1$]$= r^{2} (\sin^{2}θ +\cos^{2}θ)= r^2 (1)$Therefore,$x^{2}+y^{2}+z^{2}$isindependentof$\theta$and$\phi$.
Hence, the correct option is (a).