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 objective of this chapter is to discuss a very important issue of the effect of finite sampling with respect to either the finite length of the record or the finite sampling intervals. A few sampling theorems are discussed.

Here, we add damping to the harmonic oscillator, and explore the role of the resulting new time scale in the solutions to the equations of motion.Specifically, the ratio of damping to oscillatory time scale can be used to identify very different regimes of motion: under-, critically-, and over-damped.Then driving forces are added, we consider the effect those have on the different flavors of forcing already in place.The main physical example (beyond springs attached to masses in dashpots) is electrical, sinusoidally driven RLC circuits provide a nice, experimentally accessible test case.On the mathematical side, the chapter serves as a thinly-veiled introduction to Fourier series and the Fourier transform.

An exploration of the wave equation and its solutions in three dimensions.The chapter's mathematical focus is on vector calculus, enough to understand and appreciate the harmonic functions that make up the static solutions to the wave equation.