What characteristic indicates a solution in cylindrical symmetry?

A solution is said to exhibit cylindrical symmetry when it remains invariant under rotations about a specific axis, typically the z-axis in a cylindrical coordinate system. This means that the physical properties of the system depend only on the radial distance from the z-axis and the height along that axis, but not on the angular coordinate (which indicates the direction around the z-axis).

The characteristic of dependence on radial and axial coordinates directly reflects this symmetry. In a cylindrical symmetric solution, while the angular variable can often be disregarded or treated as constant, the system can still vary with changes in distance from the axis (radial coordinate) and height along the axis (axial coordinate). Hence, the dependence on both radial and axial coordinates correctly describes a system where the shape and behavior remain consistent as you rotate around the axis.

Other options indicate varying degrees of symmetry or lack thereof, such as uniformity or variability along the z-axis or dependence solely on the angular variable. However, the essence of cylindrical symmetry lies in the combination of radial and axial dependence, which maintains the system's characteristics across different orientations about the axis. Therefore, the correct answer captures the fundamental nature of a solution demonstrating cylindrical symmetry.