clearly by Tai [ 192, who also notes that if the integral forms are used, then their validity across regions of discontinuity should be stated as part of the postulate We have decided to use the point form in this text. In doing so we follow a long history begun by Hertz in 1890[85 when he wrote down Maxwell's differential equations as a set of axioms, recognizing the equations as the launching point for the theory of electromagnetism. Also, by postulating Maxwell's equations in point form we can take full advantage of modern developments in the theory of partial differential equations; in particular, the idea of a "well-posed" theory determines what sort of information must be specified to make the postulate useful. We must also decide which form of Maxwells differential equations to use as the basis of our postulate. There are several competing forms, each differing on the manner in which materials are considered. The oldest and most widely used form was suggeste by Minkowski in 1908 [130. In the Minkowski form the differential equations contain no mention of the materials supporting the fields; all information about material media is relegated to the constitutive relationships. This places simplicity of the differential equations above intuitive understanding of the behavior of fields in materials. We choose the Maxwell-Minkowski form as the basis of our postulate, primarily for ease of ma- nipulation. But we also recognize the value of other versions of Maxwells equations We shall present the basic ideas behind the Boffi form, which places some information about materials into the differential equations(although constitutive relationships are still required). Missing, however, is any information regarding the velocity of a moving medium. By using the polarization and magnetization vectors P and M rather than the fields D and H, it is sometimes easier to visualize the meaning of the field vectors and to understand(or predict) the nature of the constitutive relations. The Chu and Amperian forms of Maxwells equations have been promoted as useful alternatives to the Minkowski and Boffi forms. These include explicit information about the velocity of a moving material, and differ somewhat from the boffi form in the physical interpretation of the electric and magnetic properties of matter. Although each of these models matter in terms of charged particles immersed in free space, magnetization in the Boffi and Amperian forms arises from electric current loops, while the Chu form employs nagnetic dipoles. In all three forms polarization is modeled using electric dipoles. For a detailed discussion of the Chu and Amperian forms, the reader should consult the work of Kong [101, Tai 193, Penfield and Haus [145, or Fano, Chu and Adler [70 o Importantly, all of these various forms of Maxwell's equations produce the same values the physical fields(at least external to the material where the fields are measurable) We must include several other constituents, besides the field equations, to make the ostulate complete. To form a complete field theory we need a source field, a mediating field, and a set of field differential equations. This allows us to mathematically describe the relationship between effect(the mediating field) and cause(the source field).In a well-posed postulate we must also include a set of constitutive relationships and a specification of some field relationship over a bounding surface and at an initial time. If the electromagnetic field is to have physical meaning, we must link it to some observable quantity such as force. Finally, to allow the solution of problems involving mathematical discontinuities we must specify certain boundary, or jump, " conditions. 2.1.1 The Maxwell-Minkowski equations In Maxwell,s macroscopic theory of electromagnetics, the source field consists of the ector field J(r, t)(the current density) and the scalar field p(r, t)(the charge density) @2001 by CRC Press LLCclearly by Tai [192], who also notes that if the integral forms are used, then their validity across regions of discontinuity should be stated as part of the postulate. We have decided to use the point form in this text. In doing so we follow a long history begun by Hertz in 1890 [85] when he wrote down Maxwell’s differential equations as a set of axioms, recognizing the equations as the launching point for the theory of electromagnetism. Also, by postulating Maxwell’s equations in point form we can take full advantage of modern developments in the theory of partial differential equations; in particular, the idea of a “well-posed” theory determines what sort of information must be specified to make the postulate useful. We must also decide which form of Maxwell’s differential equations to use as the basis of our postulate. There are several competing forms, each differing on the manner in which materials are considered. The oldest and most widely used form was suggested by Minkowski in 1908 [130]. In the Minkowski form the differential equations contain no mention of the materials supporting the fields; all information about material media is relegated to the constitutive relationships. This places simplicity of the differential equations above intuitive understanding of the behavior of fields in materials. We choose the Maxwell–Minkowski form as the basis of our postulate, primarily for ease of ma￾nipulation. But we also recognize the value of other versions of Maxwell’s equations. We shall present the basic ideas behind the Boffi form, which places some information about materials into the differential equations (although constitutive relationships are still required). Missing, however, is any information regarding the velocity of a moving medium. By using the polarization and magnetization vectors P and M rather than the fields D and H, it is sometimes easier to visualize the meaning of the field vectors and to understand (or predict) the nature of the constitutive relations. The Chu and Amperian forms of Maxwell’s equations have been promoted as useful alternatives to the Minkowski and Boffi forms. These include explicit information about the velocity of a moving material, and differ somewhat from the Boffi form in the physical interpretation of the electric and magnetic properties of matter. Although each of these models matter in terms of charged particles immersed in free space, magnetization in the Boffi and Amperian forms arises from electric current loops, while the Chu form employs magnetic dipoles. In all three forms polarization is modeled using electric dipoles. For a detailed discussion of the Chu and Amperian forms, the reader should consult the work of Kong [101], Tai [193], Penfield and Haus [145], or Fano, Chu and Adler [70]. Importantly, all of these various forms of Maxwell’s equations produce the same values of the physical fields (at least external to the material where the fields are measurable). We must include several other constituents, besides the field equations, to make the postulate complete. To form a complete field theory we need a source field, a mediating field, and a set of field differential equations. This allows us to mathematically describe the relationship between effect (the mediating field) and cause (the source field). In a well-posed postulate we must also include a set of constitutive relationships and a specification of some field relationship over a bounding surface and at an initial time. If the electromagnetic field is to have physical meaning, we must link it to some observable quantity such as force. Finally, to allow the solution of problems involving mathematical discontinuities we must specify certain boundary, or “jump,” conditions. 2.1.1 The Maxwell–Minkowski equations In Maxwell’s macroscopic theory of electromagnetics, the source field consists of the vector field J(r, t) (the current density) and the scalar field ρ(r, t) (the charge density)