(a) In spherical coordinates
z 2 = 3 x 2 + 3 y 2 ⟺ ρ 2 cos 2 φ = 3 ρ 2 sin 2 φ cos 2 θ + 3 ρ 2 sin 2 φ sin 2 θ = 3 ρ 2 sin 2 φ ⟺ tan 2 φ = 1 3 ⟺ tan φ = ± 1 3 ⟺ φ = π 6 or 5 π 6 \begin{aligned}
z^2=3x^2+3y^2
&\iff
\rho^2\cos^2\varphi = 3\rho^2\sin^2\varphi\cos^2\theta
+3\rho^2\sin^2\varphi\sin^2\theta
=3\rho^2\sin^2\varphi \\
&\iff
\tan^2\varphi=\frac{1}{3}
\iff
\tan\varphi=\pm\frac{1}{\sqrt{3}} \\
&\iff \varphi=\frac{\pi}{6}\text{ or }\frac{5\pi}{6}
\end{aligned} z 2 = 3 x 2 + 3 y 2 ⟺ ρ 2 cos 2 φ = 3 ρ 2 sin 2 φ cos 2 θ + 3 ρ 2 sin 2 φ sin 2 θ = 3 ρ 2 sin 2 φ ⟺ tan 2 φ = 3 1 ⟺ tan φ = ± 3 1 ⟺ φ = 6 π or 6 5 π
The surface z 2 = 3 x 2 + 3 y 2 z^2=3x^2+3y^2 z 2 = 3 x 2 + 3 y 2 z ≥ 0 z\ge 0 z ≥ 0 φ = π 6 \varphi=\frac{\pi}{6} φ = 6 π z ≤ 0 z\le 0 z ≤ 0 φ = π − π 6 = 5 π 6 \varphi=\pi-\frac{\pi}{6}=\frac{5\pi}{6} φ = π − 6 π = 6 5 π
(b) In spherical coordinates
x 2 + y 2 + ( z − 1 ) 2 = 1 ⟺ ρ 2 sin 2 φ cos 2 θ + ρ 2 sin 2 φ sin 2 θ + ( ρ cos φ − 1 ) 2 = 1 ⟺ ρ 2 sin 2 φ + ρ 2 cos 2 φ − 2 ρ cos φ = 0 ⟺ ρ 2 − 2 ρ cos φ = 0 ⟺ ρ = 2 cos φ \begin{aligned}
x^2+y^2+(z-1)^2=1
&\iff
\rho^2\sin^2\varphi\cos^2\theta
+\rho^2\sin^2\varphi\sin^2\theta
+\big(\rho\cos\varphi-1\big)^2
=1 \\
&\iff
\rho^2\sin^2\varphi
+\rho^2\cos^2\varphi-2\rho\cos\varphi
=0 \\
&\iff
\rho^2-2\rho\cos\varphi =0 \\
&\iff \rho=2\cos\varphi
\end{aligned} x 2 + y 2 + ( z − 1 ) 2 = 1 ⟺ ρ 2 sin 2 φ cos 2 θ + ρ 2 sin 2 φ sin 2 θ + ( ρ cos φ − 1 ) 2 = 1 ⟺ ρ 2 sin 2 φ + ρ 2 cos 2 φ − 2 ρ cos φ = 0 ⟺ ρ 2 − 2 ρ cos φ = 0 ⟺ ρ = 2 cos φ
(c) In spherical coordinates
x 2 + y 2 = 4 ⟺ ρ 2 sin 2 φ cos 2 θ + ρ 2 sin 2 φ sin 2 θ = 4 ⟺ ρ 2 sin 2 φ = 4 ⟺ ρ sin φ = 2 \begin{aligned}
x^2+y^2=4
&\iff
\rho^2\sin^2\varphi\cos^2\theta
+\rho^2\sin^2\varphi\sin^2\theta
=4 \\
&\iff
\rho^2\sin^2\varphi =4 \\
&\iff
\rho\sin\varphi=2
\end{aligned} x 2 + y 2 = 4 ⟺ ρ 2 sin 2 φ cos 2 θ + ρ 2 sin 2 φ sin 2 θ = 4 ⟺ ρ 2 sin 2 φ = 4 ⟺ ρ sin φ = 2
since ρ ≥ 0 \rho\ge 0 ρ ≥ 0 0 ≤ φ ≤ π 0\le\varphi\le\pi 0 ≤ φ ≤ π sin φ ≥ 0 \sin\varphi\ge 0 sin φ ≥ 0
