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

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

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

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

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

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

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

$$=r^2\times 1$$

$$=r^2$$

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