MHB Change from cartesian coordinates to cylindrical and spherical

[akkex]

Hello, 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 time1) 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)

 

[akkex]

I can do the first because from Cartesian coordinates to a) z = z in cylindrical coordinates and b) z = ρcos(Φ) in spherical coordinates. But in the other exercises i can't find a way to solve it

 

[Ackbach]

So, rectangular to cylindrical:
\begin{align*}
r^2&=x^2+y^2 \\
\theta&=\text{atan2}(y,x) \\
z&=z.
\end{align*}
This is basically rectangular to polar in 2D, but with the third dimension in the $z$ direction the same. Note that the atan2 function is the "intelligent" arctangent function - it gives you an answer in the correct quadrant. Other useful equations for conversions are the ones going the other direction:
\begin{align*}
x&= r \cos(\theta) \\
y&= r \sin(\theta)
\end{align*}

Rectangular to spherical:
\begin{align*}
\rho^2&=x^2+y^2+z^2 \\
\theta&=\text{atan2}(y,x) \\
\phi&=\arccos(z/\rho).
\end{align*}
Here I'm using the convention that $\theta$ is the azimuthal angle (the one that "curves around" the $z$ axis), and $\phi$ is the polar angle (the one that comes straight down from the positive $z$ axis). I believe this is the same convention you're using.

Again, the equations going the other direction are sometimes useful in doing conversions:

\begin{align*}
x&=\rho \cos(\theta) \sin(\phi) \\
y&=\rho \sin(\theta) \sin(\phi) \\
z&=\rho \cos(\phi).
\end{align*}

So, if you take a look at problem 2, you have the equation $z=5x^2+5y^2$. Notice the pattern (this is what mathematics is all about: noticing patterns): $5x^2+5y^2=5(x^2+y^2)$. Oh! The thing in the parentheses is just $r^2$. That gives us $z=5r^2$.

The spherical conversion is more difficult. I would simply plug in the spherical-to-rectangular equations, and use trig identities to simplify:
\begin{align*}
z&=5x^2+5y^2 \\
\rho \cos(\phi)&=5[\rho \cos(\theta) \sin(\phi)]^2 + 5[\rho \sin(\theta) \sin(\phi)]^2 \\
\rho \cos(\phi)&=5\rho^2 \cos^2(\theta) \sin^2(\phi) + 5\rho^2 \sin^2(\theta) \sin^2(\phi) \\
\cos(\phi)&=5\rho \cos^2(\theta) \sin^2(\phi) + 5\rho \sin^2(\theta) \sin^2(\phi) \\
\cos(\phi)&=5\rho\sin^2(\phi)[\cos^2(\theta) + \sin^2(\theta)] \\
\cos(\phi)&=5\rho\sin^2(\phi) \\
\csc^2(\phi) \cos(\phi)&=5\rho.
\end{align*}

The other problems will work out similarly, I expect. If you can't see some high-level pattern, then plug in and use your trig identities! Let me know how you get on.