The study of magnetic fields produced by steady currents is a full-valued physical theory which like any other physical theory employs a certain mathematics. This theory has two limiting cases in which source of the field is confined on a surface or a curve. It turns out that mathematical methods to be used in these cases are completely different and differ from from that of the main of the main part of this theory, so, magnetostatics actually consists of three distinct theories. In this work, these three theories are discussed with special attention to the case current carried by a curve. In this case the source serves as a model of thin wire carrying direct current, therefore this theory can be termed magnetostatics of thin wires. The only mathematical method used in this theory till now, is the method of Green's functions. Critical analysis of this method completed in this work, shows that application of this method to the equation for vector potential of a given current density has no foundation and application of this method yields erroneous results