Based on the Bean model with a constant critical current densityjc,novel and very precise, analytical approximations for the axial induction componentBz(r) inside and outside of a finite, axially magnetized, circular cylinder with arbitrary aspect ratio in the fully penetrated, remanent critical state are presented. Starting from any mapped field profile the possibility to determine the material parameterjcby means of aglobaloptimum fit procedure is demonstrated. Some characteristic results for melt-textured yttrium barium copper oxide material are outlined. We found thatjcdetermined by vibrating sample magnetometry (VSM) exceeds that from mapping at least by about 50&percent;. This systematic discrepancy is discussed on the basis of two independent effects resulting in a local, space-dependent critical current density: theB-dependence ofjcand material inhomogeneities. The first effect becomes important for sample radii in the1-cm-range, although theBz-profile remains axially symmetric and well fitted by the Bean model. The second one may lead to asymmetricBz-profile deformations connected with a decreasing fitting significance. For the first and second effect we estimate the considerable ratio of the absolute maximum to absolute minimum of the local, critical current density of 2 and 4, respectively, which explains the observed discrepancies between theeffectivejcfrom mapping and VSM measurements. ©1997 American Institute of Physics.