Write the equation in spherical coordinates. x2 + y2 + z2 = 49.
Solution:
Given, the cartesian equation is x2 + y2 + z2 = 49.
We have to write the equation in spherical coordinates.
By using the relation to convert cartesian to spherical coordinates,
r = ρsin𝝓
x = rcos θ = ρsin𝝓cos θ
z = ρcos𝝓
y = rsinθ = ρsin𝝓sin θ
ρ = √x2 + y2 + z2 = √r2 + z2
Now, x2 + y2 + z2 = r2 + z2
According to the question,
(ρsin𝝓cos θ)2 + (ρsin𝝓sinθ) + (ρcos𝝓)2 = 49
ρ2sin2𝝓cos2θ + ρ2sin2𝝓sin2θ + ρ2cos2𝝓 = 49
Taking out common term,
ρ2sin2𝝓(cos2θ + sin2θ) + ρ2cos2𝝓 = 49
We know, cos2θ + sin2θ = 1
ρ2sin2𝝓(1) + ρ2cos2𝝓 = 49
Taking out common term,
ρ2[sin2𝝓 + cos2𝝓] = 49
Also, sin2𝝓 + cos2𝝓 = 1,
ρ2[1] = 49
ρ2 = 49
ρ = ±7
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)
Therefore, the equation in spherical coordinates is ρ = 1/√(2sin2𝝓cos2θ - 1).
Write the equation in spherical coordinates. x2 + y2 + z2 = 49.
Summary:
The equation in spherical coordinates x2 + y2 + z2 = 49 is ρ = 1/√(2sin2𝝓cos2θ - 1).
