Write the equation in spherical coordinates. (a) 4z2 = 5x2 + 5y2 (b) x2 + 3z2 = 4

(753,#53) Write the equation A) in cylindrical coordinates B) spherical coordinates. Solution: A) in cylindrical coordinates, B) in spherical coordinates,

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

Answer: r² (sin²φ cos²θ + 3cos²φ) = 4