cylindrical, spherical coordinates conversion?

Question

Hi,
Thanks for reading my question.


the question is ,
Convert equation x^2-y^2-2z^2=4
to A ) cylindrical coordinates
B) spherical coordinates

Thanks so much for your help

Answer 1

cylindrical coordinate equations are x = r cos theta, y = r sin theta, and z = z:
x^2 - y^2 -2z^2 = 4
r^2 cos^2(theta) - r^2 sin^2(theta) - 2z^2 = 4
r^2(cos^2(theta) - sin^2(theta)) - 2z^2 = 4
r^2 cos (2theta) - 2z^2 = 4

Spherical coordinate equations are
x = rho cos (theta) sin(phi)
y = rho sin (theta) sin (phi)
z = rho cos(phi)
so
rho^2 cos^2(theta)sin^2(phi) - rho^2 sin^2(theta)sin^2(phi) -
2 rho^2 cos^2(phi) = 4

rho^2 sin^2(phi)(cos^2(theta) - sin^2(theta)) - 2hro^2cos^2(phi)=4

rho^2cos(2theta)sin^2(phi) - 2rho^2cos^2(phi) = 4

Answer 2

A)
x = r cosθ
y = r sinθ
z = z

then x² - y² - 2z² = r²cos²θ - r²sin²θ -2z² = 4
r²(cos²θ - sin²θ) - 2z² = 4
or r²cos(2θ) - 2z² = 4


B)
x = r cosθ sinφ
y = r sinθ sinφ
z = r cosφ

then x² - y² - 2z² = r²cos(2θ)sin²φ - 2r²cos²φ = 4