A system of equations for calculating the electric field in curvilinear coordinates without imposing external boundary conditions is proposed. First, the system of equations is derived using generalized functions, without imposing external boundary conditions or a coordinate system. Then, in order to demonstrate the use of the system of equations, the derived system is applied to the case of a spherical scatterer, which is then narrowed to the case of a perfect electrically conducting (PEC) sphere. It is shown that for the particular case of a PEC sphere, the resulting field expression agrees with the Stratton-Chu solution for the Maxwell's equations, confirming that the proposed method can be used to reach a general solution for the electromagnetic field scattered by a PEC sphere, which allows for the calculation of its cross section.