as just setting, and is one of several factors that must be considered, as discussed in the next section Uniqueness implies that the electromagnetic state of an isolated region of space may be determined without the knowledge of conditions outside the region. If we wish to solve Maxwells equations for that region, we need know only the source density within the region and the values of the tangential fields over the bounding surface. The effects of a complicated external world are thus reduced to the specification of surface fields. This concept has numerous applications to problems in antennas, diffraction, and guided 2.2.2 Constitutive relations We now supply a set of constitutive relations to complete the conditions for well- posedness. We generally split these relations into two sets. The first describes the relationships between the electromagnetic field quantities, and the second describes chanical interaction between the fields and resulting secondary sources. All of these relations depend on the properties of the medium supporting the electromagnetic field Material phenomena are quite diverse, and it is remarkable that the Maxwell-Minkowski equations hold for all phenomena yet discovered. All material effects, from nonlinearity to chirality to temporal dispersion, are described by the constitutive relations. The specification of constitutive relationships is required in many areas of physical science to describe the behavior of "ideal materials " mathematical models of actual materials encountered in nature. For instance, in continuum mechanics the constitutive equations describe the relationship between material motions and stress tensors 209 Truesdell and Toupin [199 give an interesting set of"guiding principles"for the con cerned scientist to use when constructing constitutive relations. These include consider- ation of consistency(with the basic conservation laws of nature), coordinate invariance (independence of coordinate system), isotropy and aeolotropy(dependence on, or inde- pendence of, orientation), just setting(constitutive parameters should lead to a unique solution), dimensional invariance(similarity), material indifference(non-dependence on the observer), and equipresence (inclusion of all relevant physical phenomena in all of the constitutive relations across disciplines) The constitutive relations generally involve a set of constitutive parameters and a set of constitutive operators. The constitutive parameters may be as simple as constants of proportionality between the fields or they may be components in a dyadic relation- ship. The constitutive operators may be linear and integro-differential in nature, or may imply some nonlinear operation on the fields. If the constitutive parameters are spa tially constant within a certain region, we term the medium homogeneous within that region. If the constitutive parameters vary spatially, the medium is inhomogeneous. If the constitutive parameters are constants with time, we term the medium stationary if they are time-changing, the medium is nonstationary. If the constitutive operators involve time derivatives or integrals, the medium is said to be temporally dispersive: if space derivatives or integrals are involved, the medium is spatially dispersive. Examples of all these effects can be found in common materials. It is important to note that the constitutive parameters may depend on other physical properties of the material, such as temperature, mechanical stress, and isomeric state, just as the mechanical constitu tive parameters of a material may depend on the electromagnetic properties(principle of equipresence) Many effects produced by linear constitutive operators, such as those associated with @2001 by CRC Press LLCas just setting, and is one of several factors that must be considered, as discussed in the next section. Uniqueness implies that the electromagnetic state of an isolated region of space may be determined without the knowledge of conditions outside the region. If we wish to solve Maxwell’s equations for that region, we need know only the source density within the region and the values of the tangential fields over the bounding surface. The effects of a complicated external world are thus reduced to the specification of surface fields. This concept has numerous applications to problems in antennas, diffraction, and guided waves. 2.2.2 Constitutive relations We now supply a set of constitutive relations to complete the conditions for well￾posedness. We generally split these relations into two sets. The first describes the relationships between the electromagnetic field quantities, and the second describes me￾chanical interaction between the fields and resulting secondary sources. All of these relations depend on the properties of the medium supporting the electromagnetic field. Material phenomena are quite diverse, and it is remarkable that the Maxwell–Minkowski equations hold for all phenomena yet discovered. All material effects, from nonlinearity to chirality to temporal dispersion, are described by the constitutive relations. The specification of constitutive relationships is required in many areas of physical science to describe the behavior of “ideal materials”: mathematical models of actual materials encountered in nature. For instance, in continuum mechanics the constitutive equations describe the relationship between material motions and stress tensors [209]. Truesdell and Toupin [199] give an interesting set of “guiding principles” for the con￾cerned scientist to use when constructing constitutive relations. These include consider￾ation of consistency (with the basic conservation laws of nature), coordinate invariance (independence of coordinate system), isotropy and aeolotropy (dependence on, or inde￾pendence of, orientation), just setting (constitutive parameters should lead to a unique solution), dimensional invariance (similarity), material indifference (non-dependence on the observer), and equipresence (inclusion of all relevant physical phenomena in all of the constitutive relations across disciplines). The constitutive relations generally involve a set of constitutive parameters and a set of constitutive operators. The constitutive parameters may be as simple as constants of proportionality between the fields or they may be components in a dyadic relation￾ship. The constitutive operators may be linear and integro-differential in nature, or may imply some nonlinear operation on the fields. If the constitutive parameters are spa￾tially constant within a certain region, we term the medium homogeneous within that region. If the constitutive parameters vary spatially, the medium is inhomogeneous. If the constitutive parameters are constants with time, we term the medium stationary; if they are time-changing, the medium is nonstationary. If the constitutive operators involve time derivatives or integrals, the medium is said to be temporally dispersive; if space derivatives or integrals are involved, the medium is spatially dispersive. Examples of all these effects can be found in common materials. It is important to note that the constitutive parameters may depend on other physical properties of the material, such as temperature, mechanical stress, and isomeric state, just as the mechanical constitu￾tive parameters of a material may depend on the electromagnetic properties (principle of equipresence). Many effects produced by linear constitutive operators, such as those associated with