Beginner in mathematics, i'm studying alone.

Question

i study alone and i'm having difficulties with these issues, please help me!

Questions:

1. Determine which type of central grid is from its curves. $x ^ 2 / a ^ 2 + y ^ 2 / b ^ 2 - z ^ 2 / c ^ 2 = -1$

2. Prove that rotations in $R ^ 2$ commute. $R_a * R_b = R_b * R_a$

3. Prove that rotations in $R^3$ do not commute.

4. How to transform cylindrical coordinates into Cartesian coordinates?

5. $ρ \sin ϕ = 2$

6. A surface in $R ^ 3$ is given by coordinate $(ρ, θ, ϕ)$, how to translate into Cartesian coordinates and what is this surface?

Answer 1

This is an attempt to answer the issues:

1. Determine which type of central grid is from its curves. $x ^ 2 / a ^ 2 + y ^ 2 / b ^ 2 - z ^ 2 / c ^ 2 = -1$

This is a Two Sheeted Hyperboloid

2. Prove that rotations in $R ^ 2$ commute. $R_a * R_b = R_b * R_a$

2D Rotation Matrices are just orthonormal matrices or order 2x2, which in particular commute.

3. Prove that rotations in $R^3$ do not commute.

3D Rotation Matrices are orthonormal matrices of order 3x3, which do not commute in $R^3$ neither in general. Just provide a counterexample rotating x then y, and y then x.

4. How to transform cylindrical coordinates into cartesian coordinates?

Cylindrical Cordinates are just given by: $x=\rho \cos(\theta)$, $y=\rho \sin(\theta)$, $z=h$

5. $ρ \sin ϕ = 2$

This is simply a curve having the $\rho\sin(\theta)$ segment constant. In polar plot coordinates, this is simply the line $y=2$.

6. A surface in $R ^ 3$ is given by coordinate $(ρ, θ, ϕ)$, how to translate into cartesian coordinates and what is this surface?

Spherical Coordinates into cartesian are doing by $x=r\cos(\theta)\sin(\phi)$,$y=r\sin(\theta)\sin(\phi)$,$z=r\cos(\phi)$ (which is not the selection i prefer, but the explained in Wolfram Alpha).

Without the actual coordinates, this object can be anything.