Question

Describe the geometric meaning of the following mappings in spherical coordinates.

(a) (rho, θ, ϕ) ↦ (rho, θ + π, ϕ)
(i)-There is a rotation by π around the y-axis.
(ii)-There is a rotation by π/2 about the z-axis.
(iii)-There is a rotation by π around the z-axis.
(iv)-There is a rotation by π around the x-axis.
(v)-There is a rotation by π/2 about the y-axis.

(b) (rho, θ, ϕ) ↦ (rho, θ, π − ϕ)
(i)-There is a rotation by π around the y-axis.
(ii)-There is a reflection across the xz plane.
(iii)-There is a reflection across the yz.
(iv)-There is a rotation by π around the x-axis.
(v)-There is a reflection across the xy plane.

(c) (rho, θ, ϕ) ↦ (2rho, θ + π/2 , ϕ)
(i)-There is a rotation by π 2 about the y-axis with a radial expansion by a factor of 2.
(ii)-There is a rotation by π 2 about the z-axis with a radial expansion by a factor of 2.
(iii)-There is a rotation by π about the z-axis.
(iv)-There is a rotation by π about the y-axis.
(v)-There is a rotation by π/2 about the x-axis with a radial compression by a factor of 1/2 .

Answer 1

A. (iv)-There is a rotation by π around the x-axis.

B. (v)-There is a reflection across the xy plane.

C. None of the above, There is a rotation by π/2 about the x-axis with a radial expansion by a factor of 2.

Explanation:

A. In spherical coordinates, for some point in R³, the angle θ is designed to be the angle between the x-axis and the projection of the line that goes from (0,0,0) to that point into the xy plane.

So, if you add π to the angle it only rotate 180 degrees or π units about the x-axis

B. In spherical coordinates, for some point in R³, the angle ϕ is designed to be the angle between the z-axis and the diagonal from (0,0,0) to that point.

So, if you subtract π to the angle it would flip the diagonal in the negative part of z (assuming ϕ<π/2, otherwise, originally the diagonal would be in the negative part of z and it would flip to the positive part of z) and this could be interpreted as a reflection around the xy plane. Notice if ϕ=π/2, then π − ϕ=π/2 so everything stays the same for points that belong to xy plane (i.e that ϕ=π/2)

C. In spherical coordinates, for some point in R³, ρ is designed to be the length of the diagonal from (0,0,0) to that point. So if we multiply ρ by 2, we are expanding the length of the diagonal.

But as we´ve seen before, adding something to θ means a rotation about x-axis by what we are adding (π/2)

Then the answer is that there is a rotation by π/2 about the x-axis with a radial expansion by a factor of 2.