Chapter 8 focuses on steady open-channel flow. As opposed to pressurized flow in closed conduits, open-channel flows convey water by gravity in human-made channels and natural waterways. The cross-sectional area of open channels varies with discharge as described in Section 8.1. Section 8.2 examines resistance to flow to define the normal depth in Section 8.3 and shear stress in Section 8.4.

Chapter 2 reviews the motion of water in pipes. This chapter explains resistance to flow and major friction losses in Section 2.1, and minor head losses in Section 2.2. Head losses are combined with conservation of mass for the analysis of pipe branches and networks in Section 2.3.

Chapter 6 delves into the compressibility effects of water in pipes. Section 6.1 presents important knowledge on water compressibility. It is followed with the celerity of wave propagation in pipes in Section 6.2. The concept of water hammer is detailed in Section 6.3 with prevention measures like surge tanks in Section 6.4.

Chapter 10 delineates backwater curves and gradually varied open-channel flows. Gradually varied flows change slowly in the downstream direction. The main equation in Section 10.1 leads to the classification of water-surface profiles in Section 10.2 with calculations in Section 10.3. Energy losses at bridge crossings are covered in Section 10.4 with numerical models introduced in Section 10.5.

Chapter 12 copes with more complex flows through culverts. Culverts are described in Section 12.1 followed with an analysis of culvert performance curves in Section 12.2 and outlet works in Section 12.3.

Chapter 1 describes the physical properties of water and hydrostatics. It covers water properties, unit conversions and forces on dams. Fundamental dimensions, units and water properties are reviewed in Section 1.1. The concept of pressure and piezometric head in Section 1.2 is expanded into hydrostatic forces on plane surfaces in Section 1.3 and dams in Section 1.4.

In 1921 Weyl studied the invariance properties of quantum mechanics. In particular, he noted that the absolute phase of the wave function is not observable. In a formal and generic way, we consider a global transformation of the phase of a Dirac spinor. This can be expressed with the following unitary global gauge transformation 𝑈(𝛼)

Quantum field theory is complex. Ask ourselves why we bother with all these quantum fields in the first place? QED is a field theory of well-defined perturbation expansion and in principle any physical prediction can be calculated with practically infinite precision. So, in this chapter we explore the techniques associated with computing “higher-order” or “purely quantum” effects of electromagnetism. The 𝑆-matrix was written as the Dyson expansion (see Section 9.2), where the factor in the expansion is the electric elementary charge 𝑒.

The classical period of Roman law is conventionally taken to have ended in ad 235 with the death of the Emperor Severus Alexander. It is true that the line of independent classical jurists breaks off there. But this was not a collapse but a change of direction. The leading jurists increasingly became involved in the process of imperial law-making; and their works were the constitutions they composed in the name of their emperor. The constitutions of Diocletian in particular (ad 284–305) show that half a century after the end of the classical period the standards of classical jurisprudence had been maintained. But this was not a period in which new original juristic work appeared; instead, the trend was towards the production of anthologies or epitomes of leading classical works. It therefore seems appropriate to refer to the period from about ad 235 to 305 as the ‘epiclassical’ period of Roman law and to date the decisive break between the classical and the post-classical to about ad 300 (Wieacker 1971).