In this chapter, the Green’s function method is developed that shows how boundary values, initial conditions, and inhomogeneous terms in partial-differential equations act as source terms for response throughout a domain. The Green’s function of a given partial-differential equations is the response from an impulsive point source and satisfies homogeneous versions of whatever boundary conditions the actual response satisfies. The Green’s function propagates a response from source points to receiver points. After developing this method for the scalar wave and diffusion equations and obtaining the Green’s functions of these equations in infinite domains, the focus turns to the Green’s function method for the multitude of vectorial continuum responses governed by equations derived in Part I of the book. In particular, elastodynamics, elastostatics, slow viscous flow, and continuum electromagnetics are analyzed using the Green’s function method. The so-called Green’s tensors for each of these continuum applications in an infinite domain are obtained using the Fourier transform and contour integration.

The Fourier transform pair is derived and various conventions in its definition discussed. It is shown how to obtain forward and inverse Fourier transforms for specific functions, which results in the completeness relation being formally proven. The basic properties of the Fourier transform are derived which include the symmetry properties of the real and imaginary parts, the shifting property, the stretching property, the differentiation property, Parseval’s theorem, the convolution theorem, and the integral-moment relations. The Fourier transform pair is then used to derive the two most important theorems of probability theory: the central-limit theorem and the law of large numbers. The Fourier transform is then used to solve various initial-value problems involving the diffusion and wave equation. The chapter concludes with the way Fourier analysis is key to performing time-series analysis of recorded data, which includes both filtering of the data and topics related to the data being recorded at discrete time intervals.

In this first chapter of Part II of the book on the mathematical methods of continuum physics, the continuum governing equations in Part I are related to three simple partial-differential equations that are analyzed throughout Part II: (1) the scalar wave equation, (2) the scalar diffusion equation, and (3) the scalar Poisson (or Laplace) equation. The nature of the boundary and initial conditions required in specifying well-posed boundary-value problems for each type of partial-differential equation is derived. The three types of equations are then solved using the method of separation of variables. In so doing, the most essential things to remember about the nature of the solution to wave, diffusion, and potential boundary-value problems are presented.

The same volume-averaging procedure used in Chapter 2 shows how to transition from the Maxwell’s equations controlling the electromagnetic fields of fundamental particles in vacuum to the continuum form of Maxwell’s equations describing the electromagnetic fields averaged over large numbers of molecules. The Maxwell stress tensor is derived for the body forces acting on the molecules. The macroscopic form of Maxwell’s equations and the associated electromagnetic fields are obtained when the frame of reference is moving with the center of mass of each collection of molecules. The laws of reversible polarization are obtained by time differentiation of the electromagnetic energy density. The law of electromigration (Ohm’s law) is obtained from a nonequilibrium thermodynamics perspective. Conditions are obtained for the neglect of the material movement in the continuum theory of electromagnetism. Electromagnetic continuity conditions are derived and used on example problems. The continuum form of Newtonian gravity is derived. Expressions for the Coriolis and centrifugal forces are derived when the frame of reference is rotating about an axis.