  The local character of self-gravity along with the number of spatial
dimensions are critical issues when computing the potential and forces inside
massive systems like stars and disks. This appears from the discretisation
scale where each cell of the numerical grid is a self-interacting body in
itself. There is apparently no closed-form expression yet giving the potential
of a three-dimensional homogeneous cylindrical or spherical cell, in contrast
with the Cartesian case. By using Green's theorem, we show that the potential
integral for such polar-type 3D sectors -- initially, a volume integral with
singular kernel -- can be converted into a regular line-integral running over
the lateral contour, thereby generalising a formula already known under axial
symmetry. It therefore is a step towards the obtention of another
potential/density pair. The new kernel is a finite function of the cell's shape
(with the simplest form in cylindrical geometry), and mixes incomplete elliptic
integrals, inverse trigonometric and hyperbolic functions. The contour integral
is easy to compute; it is valid in the whole physical space, exterior and
interior to the sector itself and works in fact for a wide variety of shapes of
astrophysical interest (e.g. sectors of tori or flared discs). This result is
suited to easily providing reference solutions, and to reconstructing potential
and forces in inhomogeneous systems by superposition. The contour integrals for
the 3 components of the acceleration vector are explicitely given.
