The cylindrical coordinates of a point P are (r, θ, z) where (r, θ) are polar coordinates for (x, y). The spherical coordinates of a point P are (rho, θ, φ) where rho is the radial distance from P to the origin, θ is the same polar angle as in cylindrical/polar coordinates, and φ is the azimuthal angle, measured from the positive z axis to the ray from the origin to P. Study the picture below:

(a) Find an expression for the cylindrical coordinate r in terms of the spherical coordinates rho, θ, and φ.

r =

(b) Find an expression for the rectangular coordinate x in terms of the spherical coordinates rho, θ, and φ.

x =

(c) Find an expression for the rectangular coordinate y in terms of the spherical coordinates rho, θ, and φ.

y =

(d) Find an expression for the rectangular/cylindrical coordinate z in terms of the spherical coordinates rho, θ, and φ.

z =

(e) Find an expression for a spherical coordinate rho in terms of the rectangular coordinates x, y, and z.

rho =

(f) Find an expression for a spherical coordinate rho in terms of the cylindrical coordinates r, θ, and z.

rho =

(g) Find an expression for a spherical coordinate φ in terms of the cylindrical coordinates r, θ, and z. Careful, it should be piecewise defined.

φ =

(h) Find an expression for a spherical coordinate φ in terms of the rectangular coordinates x, y, and z. Careful, it should be piecewise defined.

φ =

(i) Find an expression for a spherical/cylindrical coordinate θ in terms of rectangular coordinates x, y, and z. Careful, it should be piecewise defined.

θ =

(j) Determine cylindrical coordinates for the point whose rectangular coordinates are (−1, √ 3, −2).

(r, θ, z) =

(k) Determine spherical coordinates for the point whose rectangular coordinates are (−1, √ 3, −2).

(rho, θ, φ) =