Question

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

Answer 1

Answer:

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.

Step-by-step 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.