Write The Equation In Spherical Coordinates. (A) X2 + Y2 + Z2 - 81 = 0 (B) X2 - Y2 - Z2 - 1 = 0
Solution:
To convert cartesian into spherical coordinates we have to rely on the following relations:
r = ρsin𝝓 x = rcos θ = ρsin𝝓cos θ
z = ρcos𝝓 y = rsinθ = ρsin𝝓sin θ
The diagram from which the above relationships are derived is:
ρ = \(\sqrt{x^{2} + y^{2} + z^{2} } = \sqrt{r^{2} + z^{2}}\)
A.We use the relationships above to substitute for x, y, z and we get
ρ2sin2𝝓cos2θ + ρ2sin2𝝓sin2θ + ρ2cos2𝝓 = 81
ρ2sin2𝝓(cos2θ + sin2θ) + ρ2cos2𝝓 = 81
We know the fundamental relationship cos2θ + sin2θ = 1
ρ2sin2𝝓(1) + ρ2cos2𝝓 = 81
ρ2[sin2𝝓 + cos2𝝓] = 81
Again sin2𝝓 + cos2𝝓 = 1, therefore
ρ2[1] = 81
ρ2 = 81
ρ = ± 9
(B) X2 - Y2 - Z2 - 1 = 0
r = ρsin𝝓 x = rcos θ = ρsin𝝓cos θ
z = ρcos𝝓 y = rsin θ = ρsin𝝓sin θ
X2 - Y2 - Z2 = 1
( ρsin𝝓cos θ )2 - (ρsin𝝓sin θ)2 - (ρcos𝝓 )2 =1
ρ2sin2𝝓cos2θ - ρ2sin2𝝓sin2θ - ρ2cos2𝝓 = 1
ρ2sin2𝝓(cos2θ - sin2θ) - ρ2cos2𝝓 = 1
ρ2sin2𝝓(cos2θ - (1 - cos2θ)) - ρ2cos2𝝓 = 1
ρ2sin2𝝓(2cos2θ -1) - ρ2cos2𝝓 =1
2ρ2sin2𝝓cos2θ - ρ2sin2𝝓 - ρ2cos2𝝓 = 1
2ρ2sin2𝝓cos2θ - ρ2[sin2𝝓 + cos2𝝓] = 1
Since sin2𝝓 + cos2𝝓 = 1
2ρ2sin2𝝓cos2θ - ρ2[1] = 1
ρ2(2sin2𝝓cos2θ - 1) = 1
ρ2 = 1/(2sin2𝝓cos2θ - 1)
ρ = 1/√(2sin2𝝓cos2θ - 1)
Write The Equation In Spherical Coordinates. (A) X2 + Y2 + Z2 - 81=0 (B) X2 - Y2 - Z2 - 1 = 0
Summary:
The equations (A) X2 + Y2 + Z2 - 8 1 = 0 (B) X2 - Y2 - Z2 - 1 = 0 are represented in spherical coordinates as ρ = ± 9 and ρ = 1/√(2sin2𝝓cos2 θ - 1) respectively.