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A practicable method for computing exciting fields is presented. To avoid volume integration and meshing one uses a transformation of Biot-Savarts's law to the coil surface. In doing so, the exciting field strength is represented by a linear combination of single- and double-layer potentials. Both can be evaluated by the fast multipole method. The method is valid for a considerably class of coil geometries including race track and bedstead coils with arbitrary cross section.