Describe the geometric meaning of the following mappings in cylindrical coordinates. (a) (r, θ, z) ↦ (r, θ, −z) There is a reflection across the xz plane. There is a reflection across the yz plane. There is a reflection across the xy plane. There is a rotation by π about the z-axis. There is a rotation by π about the y-axis. (b) (r, θ, z) ↦ (r, θ + π, −z) There is a reflection across the yz plane with a rotation by −π about the z-axis. There is a reflection across the xy plane with a rotation by −π about the y-axis. There is a reflection across the yz plane with a rotation by π about the y-axis. There is a reflection across the xz plane with a rotation by π about the z-axis. There is a reflection across the xy plane with a rotation by π about the z-axis. (c) (r, θ, z) ↦ −r, θ − π 4 , z There is a rotation by − 5π 4 about the x-axis. There is a reflection across the yz plane with a rotation by − 5π 4 about the z-axis. There is a rotation by − 5π 4 about the z-axis. There is a reflection across the yz plane with a rotation by − 5π 4 about the x-axis. There is a reflection across the xy plane with a rotation by − 5π 4 about the z-axis.