If x = r sin θ cos ϕ,
Question:

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

(a) θ, ϕ

(b) r, θ

(c) r, ϕ

(d) r.

Solution:

(a) θ, ϕ

We have:

x = r sin θ cos ϕ  ,  y = r sin θ sin ϕ and r cos θ,

$\therefore x^{2}+y^{2}+z^{2}$

$=(r \sin \theta \cos \phi)^{2}+(r \sin \theta \sin \phi)^{2}+(r \cos \theta)^{2}$

$=r^{2} \sin ^{2} \theta \cos ^{2} \phi+r^{2} \sin ^{2} \theta \sin ^{2} \phi+r^{2} \cos ^{2} \theta$

$=r^{2} \sin ^{2} \theta\left(\cos ^{2} \phi+\sin ^{2} \phi\right)+r^{2} \cos ^{2} \theta$

$=r^{2} \sin ^{2} \theta \times 1+r^{2} \cos ^{2} \theta$

$=r^{2} \sin ^{2} \theta+r^{2} \cos ^{2} \theta$

$=r^{2}\left(\sin ^{2} \theta+\cos ^{2} \theta\right)$

$=r^{2} \times 1$

$=r^{2}$

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

 

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