Write the equation in spherical coordinates. (a) 4z2 = 5x2 + 5y2 (b) x2 + 3z2 = 4
Accepted Solution
A:
Refer to the diagram shown below.
The transformation from the Cartesian coordinate system (x,y,z) to the spherical coordinate system (r,θ,φ) is given by x = rsinφcosθ y = rsinφsinθ z = rcosφ
Part a. The given equation is 4z² = 5x² + 5y². In spherical coordinates, obtain 4r²cos²φ = 5r²sin²φ cos²θ + 5r²sin²φ sin²θ = 5r²sin²φ(cos²θ + sin²θ) = 5r²sin²φ 4/5 = tan²φ tanφ = 2/√5 = (2/5)√5
Answer: tanφ = (2/5)√5
Part b. The given equation is x² + 3z² = 4 In spherical coordinates, obtain r²sin²φ cos²θ + 3r²cos²φ = 4