We present a method for calculating analytical expressions for the velocity, magnetic field, electric field, and current density arising from the quasi-static stressing of an initially potential, inhomogeneous coronal magnetic field by photospheric flows. Four exact solutions that are produced by a wide variety of driving flows are examined. We have found that large gradients in the vorticity can lead to the generation of strong current layers, across which the magnetic field is continuous. In two of the exact solutions these appear as logarithmic singularities in the current density and first spatial derivatives of the vorticity. Two other solutions, on the other hand, driven by continuous and differentiable photospheric flows lacking large gradients in the vorticity, exhibit little intensification of the current density. When current layers appear, they are located at the quasi-separatrix layers (QSL) of the magnetic configuration, where small displacements of the magnetic field footpoints at one part of the photosphere lead to large displacements of the footpoints at the other end of the field line. This extends previous QSL work by providing an additional constraint on the driving flow in order that strong currents are generated at such layers in the course of a quasi-static evolution of the magnetic field. ¿ 1997 American Geophysical Union