Change from cartesian coordinates to cylindrical and spherical

Question

I have 6 equations in Cartesian coordinates a) change to cylindrical coordinates b) change to spherical coordinate This book show me the answers but i don't find it If anyone can help me i will appreciate so much! Thanks for your time

1) z = 2 a) z = 2 b)ρcos(Φ) = 2

2) z = 5x² + 5y² a) z = 5r² b)5ρ = cos(Φ)cosec²(Φ)

3) x² + y² + z² = 9 a) r² + z² = 9 b)ρ = 3

4) x² + y² + 2z² = 4 a) r² +2z² = 4 b)ρ² (1 + cos²(Φ) = 4

5) x² - y² -2z² = 1 a) 2z² = r²cos(2θ) b)ρ² (sin²(Φ)cos(2θ) -2cos²(Φ) = 1

6) x² + y² = 2x a) r = 2cos(θ) b)ρsin(Φ) = 2cos(θ)

Answer 1

The conversion from Cartesian to cylindrical coordinates reads $$ x=r\cos(\theta),\quad y=r\sin(\theta),\quad z=z\,, $$ and from Cartesian to spherical coordinates $$ x=\rho\sin(\phi)\cos(\theta),\quad y=\rho\sin(\phi)\sin(\theta),\quad z=\rho\cos(\phi)\,. $$ Inserting this into the equations 1) - 6) should give you the posted solutions a) and b) for each case.