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.

The extensive thermodynamic variables of a fluid are introduced as the internal energy, volume, and number of molecules. The entropy is defined and also shown to be extensive. Taking the total derivative of the internal energy produces the first law of thermodynamics and defines the intensive parameters of temperature, pressure, and chemical potential. Changing variables from extensive variables to intensive variables is accomplished with the Legendre transform and defines alternative energies such as the Helmholtz free energy, enthalpy, and Gibbs free energy. Thermodynamic equilibrium requires that each element of a system have the same temperature, pressure, and chemical potential. For equilibrium to be stable, the material properties of each element must satisfy certain derived constraints. First-order phase transition are treated for a single-species system. Multispecies systems are treated and a widely used expression for how the chemical potentials of each species depend on the concentration of the species is derived. Chemical reactions are treated as is osmosis. The thermodynamics of solid systems is addressed along with mineral solubility in liquid solutions.

In this chapter, we derive Sturm–Liouville theory that introduces a broad class of eigenfunctions that are convenient to use for representing functions. Sturm–Liouville theory provides the basis of the Fourier-series method of representing functions that is the main focus of the chapter and that also is the foundation of Fourier analysis. We show how to calculate Fourier series and to use Fourier series to obtain the solution of boundary-value problems posed in Cartesian coordinates. It is seen that the main advantage of an eigenfunction approach for solving boundary-value problems is that either the inhomogeneous source term in the differential equation or the boundary values may be time dependent, which they cannot be in the method of separation of variables.

The primary task of electrostatics is to find the electric field of a given stationary charge distribution. In principle, this purpose is accomplished by Coulomb’s law, in the form of Eq. 2.8:

The fundamental problem electrodynamics hopes to solve is this (Fig. 2.1): We have some electric charges, $q_1,q_2,q_3,\dots$ (call them source charges); what force do they exert on another charge, $Q$ (call it the test charge)? The positions of the source charges are given (as functions of time); the trajectory of the test particle is to be calculated.

Remember the basic problem of classical electrodynamics: we have a collection of charges $q_1,q_2,q_3,\dots$ (the “source” charges), and we want to calculate the force they exert on some other charge $Q$ (the “test” charge – Fig. 2.1). According to the principle of superposition, it is sufficient to find the force of a single source charge – the total is then the vector sum of all the individual forces.