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.

Nonequilibrium transport equations are derived for two types of diffusive systems: (1) viscous fluids made of a single molecular species that support thermal flux and (2) two-component (solute and solvent) miscible fluids that support solute flux and thermal flux. The general statement of energy conservation for any viscous fluid is derived and used to obtain the statement of entropy conservation for each system type. This identifies the irreversible entropy production of each system, which in turn produces linear transport laws relating the nonequilibrium diffusive flux to the gradients in the intensive parameters. The matrix of transport coefficients in the transport laws is proven to be symmetric (Onsager symmetry) using the continuum governing equations and requires the direction of flow to be reversed to obtain symmetry. Capillary physics is treated using Cahn–Hilliard theory that resolves the gradients in concentration across transition layers separating two immiscible, or partially miscible, fluid. The rules of contact-line movement (imbibition and drainage) in conduits are derived from a more macroscopic perspective where the transition layers are modeled as sharp interfaces.

This chapter shows how to transition exactly from discrete molecular dynamics to the averaged continuum dynamics controlling the movement of the center of mass of large numbers of molecules. Discrete particle dynamics is described from the classical Newton–Maxwell perspective and from the quantum perspective with an emphasis on how quantum effects control the force interactions between molecules. Representing atoms using the Dirac delta function in three dimensions (a field) is introduced along with the volume-averaging theorem that defines the macroscopic gradient of volume-averaged fields. The continuum statements of the conservation of mass and momentum of large numbers of atoms are derived. It is shown that the forces causing the center of mass of a collection of molecules to move come entirely from molecules that lie outside the collection. In so doing, the stress tensor is obtained as sums over the molecular-force interactions and a sum over the thermal (random) kinetic energy of the molecules. Body forces are defined as the long-range force fields of electromagnetism and gravity acting on each collection of molecules.

The law of Newtonian viscosity is derived and the suite of continuum equations controlling the mechanics of fluids presented. Conditions for viscous flow to be considered incompressible are derived and the Navier–Stokes equations defined. Dimensional analysis is described along with the idea of similarity of two flow fields occurring on different spatial and temporal scales. The nature of the boundary and initial conditions for a flow domain are obtained that result in unique solutions of the linear form of the Navier–Stokes equations along with the specific boundary conditions on the flow fields that hold at fluid–solid and fluid–fluid interfaces. Analytical solutions of viscous flow are obtained for a range a specific, and simple, steady-state flow geometries. Time harmonic flow in straight conduits is determined as is the magnetohydrodynamic flow taking place in straight conduits filled with an electrically conducting fluid and a magnetic field applied perpendicularly to the conduit. In the guided exercises, the lubrication approximation is used to obtain approximate solutions for a range of flow scenarios.

In this chapter, the student learns how to perform certain classes of definite integrals using contour integration methods. Although the integration variable is real for most integrals of interest, such as the inverse Fourier transform, analysis of the integral is extended to complex values of the integration variable and theorems related to integrating around closed contours on the complex plane are used to solve classes of definite integrals. The key theorems include Cauchy’s theorem for integrating so-called analytic functions, Jordan’s lemma, and the residue theorem for the important case where inside a closed contour on the complex plane, the integrand has places called singularities at which the function is not well behaved. Contour integration is used to analyze and derive results for the constitutive laws of a material when the current response depends not just on current forcing but also on the history of the forcing. This topic is called delayed linear response, which is developed at length. Contour integration, when combined with Fourier transforms, provides the solution of various types of initial-value and boundary-value problems in infinite and semi infinite domains.

The rules of macroscopic elastic response are derived in an exact way by first stating the time rate at which mechanical work is performed in deforming a collection of molecules, which is the time rate at which internal elastic energy is being reversibly stored in the molecular bonds. From this work rate, the definition of the average stress tensor is obtained as well as the exact statement of the strain rate. An additional time derivative of the average stress tensor then gives Hooke’s law in its most general nonlinear form. How the elastic stiffnesses in Hooke’s law change with changing strain is derived. Displacement is defined and the shape change and volume change of a sample are understood through how the displacements of the surface bounding the sample are related to the strain tensor. Elastodynamic plane body-wave response is obtained, as is reflection and refraction of plane body waves from an interface and evanescent surface waves. It is shown how sources of elastodynamic waves such as cracking and explosions are represented as equivalent body forces.

This chapter is meant to be a student’s first introduction to tensors. Self-contained and complete, the student learns how tensors are defined, written, and used. The scalar and vector products are defined along with the physical meaning of the divergence and curl differential operations that act on tensors of any order. The integro-differential theorems are introduced in three dimensions, which include the fundamental theorem of calculus in three dimensions, Stokes’ theorem and the Reynolds’ transport theorem. The student learns how to derive a long list of tensor-calculus product rules that are valid in any coordinate system. The Taylor series in three-dimensional space is derived, which involves tensors of all orders. Functions of second-order tensors are defined. Isotropic tensors of all tensorial orders are obtained and used in proving Curie’s principle for the constitutive laws in an isotropic material. Tensor calculus in orthogonal curvilinear coordinates is developed. Finally, the Dirac delta function is introduced along with its integral and differential properties and uses.