MRI as a medical diagnostics tool is still unavailable to the majority of the developing world. Therefore the design and development of new low-cost hardware are essential. The design of gradient coils corresponding to this hardware is necessary for conventional imaging and reconstruction methods to be used. The target field method, which was originally developed to deal with longitudinal main magnetic fields, is applied to a transverse field, as produced by a Halbach permanent magnet array. Using this method current densities for gradient fields in the three spatial directions are derived. Subsequently, using stream functions, wire patterns for the three gradient coils are determined. These are verified using a commercial magneto-static solver. Furthermore, one of the gradients is constructed to validate the performance of the method. The measured fields are in good agreement with the simulations and their prescribed target fields. This confirms that the proposed method provides a reliable way to design and manufacture gradient coils for various requirements. Based on the experimental review of the constructed coil three optimized gradients are proposed for the low field MRI system developed at the LUMC in cooperation with the TU Delft. The method can also be readily generalized to other geometries and requirements due to the robust fundamental physical basis and accuracy with respect to computer simulations.