Starting from the definition of tensorial objects by their response to coordinate transformation, this chapter builds the flat space vector calculus machinery needed to understand the role of the metric and its associated geodesic curves in general. The emphasis here is on using tensors to build equations that are “generally covariant,” meaning that their content is independent of the coordinate system used to express them. Motivated by the transformation of gravitational energy sources, the gravitational field should be a second-rank tensor, and given the way in which that tensor must show up in a particle motion Lagrangian, it is natural to interpret that tensor as a metric.

State aid law controls public spending by Member States by prohibiting aid which damages the internal market and encouraging spending on projects of interest to the EU economy. The Court of Justice plays a central role in delimiting the scope of application of State aid law. The Commission has extensive powers to investigate State aid and may order recovery of funds that are granted illegally. This remedy harms the beneficiary but does little to deter the Member State granting aid. The Commission has been successful in reducing the grant of State aid and encouraging States to fund certain types of State aid which contribute to the EU’s emerging industrial policy. Moments of economic chaos like the financial crisis in 2008 and the Covid-19 pandemic led to a significant relaxation of State aid discipline but the Commission used these two crises to press for further economic integration in the form of the Banking Union and the Recovery and Resilience Facility respectively.

The multiscale nature of dispersed multiphase flows makes their characterization challenging. A single-phase flow may be reasonably characterized in terms of nondimensional parameters, such as the Reynolds number, Mach number, or Rayleigh number. But characterization of a multiphase flow requires additional parameters that describe the dispersed phase and its relation to the continuous phase. In this chapter we will introduce mathematical definitions of some basic quantities and explain how they characterize the dispersed multiphase flow.

Services are the largest part of modern economies, but often highly regulated, making cross-border service activity hard to achieve. Sometimes, as in the case of abortion, healthcare, education or gambling, they have an important social, redistributive or moral aspect which makes liberalisation of cross-border services politically sensitive. Yet the Court of Justice’s case law is very far reaching, treating any measures which hinder or make less attractive the provision of cross-border services as prohibited unless they can be justified. This applies not just to the State, but to any body restricting market access, including trade unions concerned to exclude low-cost competition from posted workers. Much of this case law has been codified in the Services Directive, which also addresses freedom of establishment, but the Directive has so many exclusions that Article 56 of the Treaty on the Functioning of the European Union and the case law remain important, as does sector-specific legislation such as that on free movement of patients.

The Euler–Euler (EE) approach derives its name from the fact that both the continuous and the dispersed phases are solved in the Eulerian frame of reference. For the fluid phase, the Eulerian frame is the natural choice and was pursued both in the particle-resolved (PR) and the Euler–Lagrange (EL) approaches. Particles are, however, inherently Lagrangian, and an Eulerian representation is possible only when the individual nature of the particles is erased. This requires that the particle-related Lagrangian quantities be suitably averaged, so that Eulerian fields of these quantities can be defined. The averaging process will allow particle volume fraction, particle velocity, and particle temperature fields to be defined as functions of space and time.

In Chapter 2, we complete our discussion of standard introductory concepts, with a focus on rationality, choice, and opportunity costs. We extend these concepts to the economics of groups, with a discussion of incentives for individuals within small versus large groups. And we apply this to a discussion of shirking and the usefulness of “tough bosses.”

Why does diplomacy exist? How does it contribute to a country’s national interest? How critical is it to national security? The answers to these questions are important as much for people in government as for those on the outside. A country’s diplomatic service is the steward of its national interests abroad. International civil servants, who work on the staff of multilateral organizations like the United Nations and the European Union, are supposed to serve global or regional interests. Both bilateral and multilateral diplomats manage and participate in the daily conduct of international relations. In carrying out their duties, they work within a diplomacy architecture–systems that have been established at the national and global levels. Before we discuss these systems, we need to understand how diplomacy relates to other key terms and concepts, such as national interest, national security and foreign policy.

In this chapter, we investigate the problem of heat transfer from an isolated rigid sphere subjected to a cross flow of different temperature. This thermal problem is analogous to the flow problem considered earlier, and the interest here is to establish an expression for heat transfer in terms of the undisturbed ambient flow, which must now be characterized both in terms of relative velocity and temperature difference. We will start our investigation with rigorous analytical results in the Stokes and the small Péclet number regime.

Economic tradecraft is a set of duties, responsibilities and skills required of diplomats working in economic affairs. It is a key instrument in the diplomatic tradecraft toolbox. As is the case with their colleagues in the political career track, economic officers work both at diplomatic missions abroad and at headquarters. On the surface, it may appear that a country’s economic and commercial diplomats do the same type of work abroad, but that is not quite the case. Economic officers inform policymaking at headquarters by monitoring and analyzing economic trends and developments in the receiving state. They also advocate for host-government policies aimed at leveling the playing field for companies from the home country and against regulations that hurt those businesses. Commercial diplomats directly help industries and individual companies in starting or expanding business and investment in the host country. Conversely, they facilitate investment by local firms in the home country.

The Court of Justice comprises two courts, the Court of Justice and the General Court. The most high-profile task of the latter relates to individuals coming before it seeking judicial review of EU Institutions. This chapter focuses on one procedure, the preliminary reference procedure, whereby national courts refer questions of EU law to the Court of Justice where these are necessary to decide disputes before the former. The Court of Justice has recrafted this procedure to establish a judicial order in which it is the apex court, with all courts having an unfettered and immediate right of reference to it. This judicial order does four things: develops EU law, secures the judicial review of EU Institutions, protects the uniformity of EU law and upholds the independence of the judiciary within EU States. Within this order, national courts against whose decisions there is no remedy must refer, as must all courts who assess that an EU measure is invalid. Other courts have a discretion to refer, with this discretion extending to all courts where a materially identically dispute has decided the point of law in question or the act is so clear it does not require interpretation.

This chapter considers Economic and Monetary Union (EMU). Nineteen States currently have the euro as their currency. EMU is built around three pillars. There is, first, an independent European Central Bank with the exclusive right to authorise the issue of the euro. It has also developed extensive powers to purchase large amounts of securities to stimulate weak economic performance in part of the euro area, and of prudential supervision over credit institutions within the euro area. States not participating in the euro are not bound by its decisions. Secondly, States commit not to incur excessive deficits, to run balanced budgets over the medium term and to correct macroeconomic imbalances. Euro area States can be subject to heavy sanctions for not meeting their obligations. Thirdly, there is coordination of domestic economic policy. The central arena for oversight of much of this is the European Semester for Economic Policy Coordination. There, the EU Institutions assess domestic economic and fiscal performance as well as performance in the fields of social policy and climate change, with recommendations to be implemented by States in their budgets in the subsequent year.