Abstract-In this paper, the independence, completeness of Maxwell's equations and uniqueness theorems in electromagnetics are reviewed. It is shown that the four Maxwell's equations are independent and complete.A complete uniqueness theorem is proposed and proven for the first time by pointing out logic mistakes in the existing proof and presenting a truth table. Therefore, electrostatics and magnetostatics can be reduced from dynamical electromagnetics in all aspects including not only the equations as subsets of Maxwell's equations but also the corresponding uniqueness theorems. It is concluded that the axiomatic system of electromagnetic theory must consist of all four Maxwell's equations.

Abstract-In this paper, the physical spline finite element method (PSFEM) is applied to the fullwave analysis of inhomogeneous waveguides. Combining (rectangular) edge element and the PSFEM, the cubic spline interpolation is successfully applied to the wave equation. For waveguide problems, the resulting nonlinear eigenvalue problem is solved by a simple iteration method in which the initial estimate is taken as the linear Lagrange interpolation, and then the solutions are improved by a few iterations. The bandwidth of the resultant matrix from the PSFEM is the same as that of linear Lagrange interpolation and is sparse. As a result, sparse matrix solver can be used. Three typical examples are demonstrated and compared with the analytical solutions and with the linear Lagrange interpolation results. It is observed that the present method converges much faster than the Lagrange interpolation method.

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Abstract-Based on a generalized Helmholtz's identity, definitions of an irrotational vector and a solenoidal vector are reviewed, and new definitions are presented. It is pointed out that the well-known uniqueness theorem of a vector function is incomplete. Although the divergence and curl are specified, for problems with finite boundary surfaces, normal components are not sufficient for uniquely determining a vector function. A complete uniqueness theorem and its two corollaries are then presented. It is proven that a vector function can be uniquely determined by specifying its divergence and curl in the problem region, its value (both normal and tangential components) on the boundary.