The macroscopic theory of the electromagnetic field is based on the following four vector quantities:

Based on empirical experience, it is reasonable to assume that these quantities are continuous and continuously differentiable almost everywhere in the computational domain, except for sets of zero measure such as interfaces separating materials with different electromagnetic properties. The points where the field is continuous are called regular, the others are singular. The electromagnetic field may be classified with respect to a number of various properties and characteristics, for example:

• field sources (electric charges, currents, permanent magnets),

• dimensionality given by the lowest number of coordinates that fully describe the field distribution (ID, 2D, 3D models),

• boundedness (fields bounded in a finite domain or open-boundary fields),

• time evolution of the field quantities [static (stationary) fields, time-harmonic fields, general time dependencies],

• types of media (homogeneous or inhomogeneous, linear or nonlinear, isotropic or anisotropic, disperse or indisperse),

• motion of sources or media, and others.

In this chapter we will need to integrate both scalar and vector fields along smooth curves. Without loss of generality, we can assume that a smooth curve C C Rd always can be parameterized from the interval (0,1), i.e.,

C = C(s) = (Cl,...,Cd)(s), -se (0,1), as shown in Figure 7.1.

For simplicity we use the same symbol C for the curve and its parameterization. A curve C is called smooth if the derivatives C[(s) = (dC¿/ds)(s) of all of its components are continuous in (0,1). Without loss of generality, we assume that the parameterization C(s) of a smooth curve C satisfies the condition

Then for every £ 6 (0,1) the derivative (dC/ds)(£) is a vector tangential to the curve C at the point C«) € Rd. A curve C is said to be closed if it is defined in [0,1] andC(0) = C( 1). A scalar field ip : Rd —> R is integrated along the curve C using the standard formula

Í ipdC = i\(C(s))\C'{s)\ds, Jc Jo where |C'(s)| is the magnitude of the derivative C'(s), ™ -

For example, the length of C is obtained by integrating the function tp(s) = 1,

Vector fields F : Rd —>■ Rd are integrated along the curve C using another standard formula,

7.1.2 Maxwell's equations in integral form

The mathematical model of the electromagnetic field, that nowadays is known as the Maxwell's equations, first appeared in the Treatise on Electricity and Magnetism by James Clerk Maxwell in 1873. These equations are assumed to be one of the greatest achievements of the 19th-century mathematics. Among Maxwell's other remarkable contributions were (a) the observation that light is an electromagnetic phenomenon (around 1862) and (b) the development of the Maxwell-Boltzmann kinetic theory of gases, which he published independently of Ludwig Boltzmann in 1866.

The Maxwell's equations consist of Ampere's law, Faraday's law of induction, and Gauss' laws for electricity and magnetism. Consider a planar simply-connected area A whose boundary C is a closed smooth curve. Ampere's law.

postulates that the line integral of the tangential component of the magnetic field strength H along C is proportional to the total current passing through the area A in the normal direction. This current is given by the sum of the conductive current I and displacement current d^/di. The conductive current J is a scalar quantity defined by where J stands for the vector-valued density of conductive currents. The dielectric flux i" is defined by

where D is the electric flux density and the symbol v stands for the unit normal vector to A, oriented positively with respect to the orientation of the curve C (right-hand rule). Faraday's law of induction, d$

Jc di represents an analogous rule for the electric field strength E: The line integral of the tangential component of the electric field E along any closed smooth planar loop C is equal to the negative of the rate of temporal change of the magnetic flux $ through the corresponding area A in the normal direction. The magnetic flux $ is defined by

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