This chapter develops the Navier–Stokes equations using a Lagrangian description. In doing so, the concept of a stress tensor and its role in the overall force balance on a fluid element is discussed. In addition, the various terms in the stress tensor as well as the individual force terms in the Navier–Stokes equations are investigated. The chapter ends with a discussion on the incompressible Navier–Stokes equations.

This chapter serves as an introduction to the concept of conservation and how conservation principles are used in fluid mechanics. The conservation principle is then applied to mass and an equation known as the continuity equation is developed. Various mathematical operations such as the dot product, the divergence, and the divergence theorem are introduced along the way. The continuity equation is discussed and the idea of an incompressible flow is introduced. Some examples using mass conservation are also given.

In this chapter, a concept known as scaling is introduced. Scaling (also known as nondimensionalization) is essentially a form of dimensional analysis. Dimensional analysis is a general term used to describe a means of analyzing a system based off the units of the problem (e.g. kilogram for mass, kelvin for temperature, meter for length, coulomb for electric change, etc.). The concepts of this chapter, while not entirely about the fluid equations per se, is arguably the most useful in understanding the various concepts of fluid mechanics. In addition, the concepts discussed within this chapter can be extended to other areas of physics, particularly areas that are heavily reliant on differential equations (which is most of physics and engineering).

In addition to the continuity equation, there is another very important equation that is often employed alongside the Navier–Stokes equations: the energy equation. The energy equation is required to fully describe compressible flows. This chapter guides the student through the development of the energy equation, which can be an intimidating equation. A discussion on diffusion and its interplay with advection is also included, leading to the idea of a boundary layer. The chapter ends with the addition of the energy equation in shear-driven and pressure-driven flows.

This chapter considers the role of essential security as a justification for breaching obligations under investment treaties. It notes the link between the protection of national security and the doctrine of necessity under customary international law. The chapter looks at some of the key national security screening instruments of various countries, notably the US FIRRMA.

This chapter introduces the concept of foreign direct investment, tracing its history and economic justifications. The chapter goes on to explain the sources of international investment law, the most significant of which are treaties.

This chapter examines political risk insurance as an alternative to international investment treaties, looking at some of the world’s leading providers of this service. It continues with a consideration of investment incentives, followed by a discussion of some of the main sources for further information on international investment law. This chapter then offers a concluding overview of some of the central debates in international investment law.

The dynamics of the EU legal order result from interactions between a great diversity of actors. The main characteristics of key actors explain the specific role attributed to each of them in the functioning of the EU, as will be further set out in subsequent chapters. The Member States, to start with, play a prominent and decisive role in shaping the nature and boundaries of the EU legal order (Section 2.1). Alongside this, nationals of the Member States are today an important part of the system of checks and balances, and are referred to as EU citizens (Section 2.2). Yet it is to the specific and sophisticated set of institutions (Section 2.3) and complementary organs (Section 2.4) that breathe life into the EU legal order that much of our attention in this chapter will be devoted.

We will study the pricing of European contingent claims within this model through determining the existence of a risk-neutral probability measure and replicating strategies. The results and computations in this chapter serve as motivation and preparation for the Black–Scholes model.

The main aim in this chapter is to introduce the Black–Scholes model and to study how this model is used to price financial options. Although, in reality, trading is done by computers and therefore stocks are traded at discrete times, the times between successive trades can be extremely short and therefore trading strategies can be well approximated by continuous-time processes. The advantage of this is that one can use the machinery of stochastic calculus to do computations which would potentially be very complicated if attempted directly using the discrete-time formulas from the first half of this book.