The current distribution in a set of parallel rings and patches, called islands, positioned at the surface of a cylinder, is investigated. The current is driven by an externally applied source current. The islands are rectangular pieces of copper (patches) placed in parallel between the rings. The eddy currents in the islands induce currents in the rings that vary in the tangential direction. From the quasi-static Maxwell equations, an integral equation for the current distribution in the strips is derived. The Galerkin method, using global basis functions, is applied to solve this integral equation. It shows fast convergence. The global basis functions are Legendre polynomials in the axial direction and 2p-periodic trigonometric functions in the tangential direction. The Legendre polynomials efficiently cope with the singularity of the kernel function of the integral equation. Explicit numerical results are shown for three configurations. Apart from the current distributions, the resistance and self-inductance of the three systems of rings and islands are computed. The resulting tool can be used to qualitatively understand eddy currents in z-gradient coils, and as such enable the incorporation of eddy currents in the optimization of gradient-coil design.