  This paper presents a new reformulated theorem for fields embedded on a
sphere or a disk. We focus in particular on the associated sphere of a disk
when closing its only one boundary. We call this the disk-sphere duality
theorem for the study of fields topological properties. For that purpose, we
use the Poincar{\'e}-Hopf theorem and the boundary number theorem to firmly
support our developments. In this context, the state of a $n$-symmetry
direction field will be analyzed to show that disk-sphere field duality is
closely related to the behavior near the disk's boundary.
