Answer to Question #335775 in Analytic Geometry for Dhanush

Question #335775

Express the following surfaces in spherical coordinates (i) yz=2. (ii) y^2+z^2-x^2=1

1
Expert's answer
2022-05-05T12:22:34-0400

1 ) "yz=2\\\\"

We know that

"y=r\\sin\u03b8\\sin\u03d5 \\\\\nz=r\\cos{\\theta}"


"yz= r\\sin\u03b8\\sin\u03d5\\ r\\cos{\\theta}\n\\\\=r^2\\sin{\\theta}\\cos{\\theta}\\cos{\\phi}\n\\\\=\\frac{1}{2}r^2\\sin{2\\theta}\\cos{\\phi}"

Now, 

"yz=2\n\\\\ \\Rightarrow \\frac{1}{2}r^2\\sin{2\\theta}\\cos{\\phi}=2\n\\\\ \\Rightarrow r^2\\sin{2\\theta}\\cos{\\phi}=4, \\ r>0,\\ \\theta\\in[0,\\pi], \\ \\phi\\in[0,2\\pi]"



2) y2+z2-x2=1

We know that

х=r sinθ cosϕ

y=r sinθ sinϕ

z=r cosθ


y2+z2-x2=r2 sin2θ sin2ϕ + r2 cos2θ - r2 sin2θ cos2ϕ = r2 sin2θ (sin2ϕ -cos2ϕ )+ r2 cos2θ= r2 sin2θ (1- 2cosϕ ) +r2 cos2θ=

=r2 sin2θ - 2r2 sin2θcosϕ +r2 cos2θ = r2 (sin2θ+ cos2θ)- 2r2 sin2θcosϕ = r2 - 2r2 sin2θcosϕ =1

r2 - 2r2 sin2θcosϕ =1 ,  r>0, θ∈[0,π], ϕ∈[0,2π]


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