Explain how to express ρ, sin(θ), cos(θ), cos(φ), sin(φ) as functions of x, y, z.

1) i) Explain how to derive the representation of the Cartesian coordinates x, y, z in terms of the spherical coordinates ρ, θ, φ to obtain

(0.1) r =< x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), z = ρcos(φ) > .

What are the conventional ranges of ρ, θ, φ?

ii) Conversely, explain how to express ρ, sin(θ), cos(θ), cos(φ), sin(φ) as functions of x, y, z.

iii) Consider the spherical coordinates ρ, θ, φ. Sketch and describe in your own words the set of all points x, y, z in x, y, z space such that:

a)0≤ρ≤1,0≤θ<2π,0≤φ≤π b)ρ=1, 0≤θ<2π,0≤φ≤π, c)0≤ρ<∞, 0≤θ<2π,φ=π, d)ρ=1, 0≤θ<2π,φ=π,

e) ρ=1,θ=π,0≤φ≤π.f)1≤ρ≤2,0≤θ<2π,π ≤φ≤π. 463

iv) In a different set of Cartesian Coordinates ρ, θ, φ sketch and describe in your own words the set of points (ρ, θ, φ) given above in each item a) to f). For example the set in a) in x, y, z space is a ball with radius 1 and center (0,0,0). However, in the Cartesian coordinates ρ, θ, φ the set in a) is a rectangular box.

2) [Computation and graphing of vector fields]. Given r =< x,y,z > and the vector Field

(0.2) F (x, y, z) = F (r) =< 1 + z, yx, y >,