Q. 75.0( 2 Votes )

# Mark the correct alternative in the following:

If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then x^{2} + y^{2} + z^{2} is independent of

A. θ, ϕ

B. r, θ

C. r, ϕ

D. r

Answer :

Given:

X = r sin θ cos ϕ

Y = r sin θ sin ϕ

Z = r cos θ

⇒ x^{2}+y^{2}+z^{2}

⇒ (r sin θ cos ϕ )^{2} + (r sin θ sin ϕ) ^{2} + (r cos θ )^{2}

⇒ r^{2}sin^{2}θcos^{2}ϕ + r^{2}sin^{2}θsin^{2}ϕ + r^{2}cos^{2}θ

⇒ r^{2}sin^{2}θ(cos^{2}ϕ+ sin^{2}ϕ) + r^{2}cos^{2}θ

⇒ r^{2}sin^{2}θ + r^{2}cos^{2}θ

⇒ r^{2}(sin^{2}θ + cos^{2}θ)

⇒ r^{2}.

∴ It is independent of θ and ϕ .

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