Take anything in the universe, put it in a box, and heat it up. Regardless of what you start with, the motion of the substance will be described by the equations of fluid mechanics. This remarkable universality is the reason why fluid mechanics is important.

The key equation of fluid mechanics is the Navier-Stokes equation. This textbook starts with the basics of fluid flows, building to the Navier-Stokes equation while explaining the physics behind the various terms and exploring the astonishingly rich landscape of solutions. The book then progresses to more advanced topics, including waves, fluid instabilities, and turbulence, before concluding by turning inwards and describing the atomic constituents of fluids. It introduces ideas of kinetic theory, including the Boltzmann equation, to explain why the collective motion of 1023 atoms is, under the right circumstances, always governed by the laws of fluid mechanics.

Take water and push it through a pipe. If the flow is slow, then everything proceeds in a nice, orderly fashion. But as you force the water to move faster and faster, it starts to wobble. And then those wobbles get bigger until, at some point the flow loses all coherence as it tumbles and turn, tripping over itself in an attempt to push forwards. This is turbulent flow.

Understanding turbulence remains one of the great outstanding questions of classical physics. Why does it occur? How does it occur? How should we characterise such turbulent flows? The purpose of this chapter is to take the first tiny steps towards addressing these questions.

The periodic table is one of the most iconic images in science. All elements are classified in groups, ranging from metals on the left that go bang when you drop them in water through to gases on the right that don’t do very much at all. The purpose of this chapter is to start to look at the periodic table from first principles, to understand the structure and patterns that lie there.

Many of the most interesting things in fluid mechanics occur because simple flows are unstable. If they get knocked a little bit, the fluid curls up into interesting shapes, or dissolves into some messy turbulent flow. In this chapter, we start to understand how these processes can happen.

Any education in theoretical physics begins with the laws of classical mechanics. The basics of the subject were laid down long ago by Galileo and Newton and are enshrined in the famous equation $F=ma$ that we all learn in school. But there is much more to the subject and, in the intervening centuries, the laws of classical mechanics were reformulated to emphasise deeper concepts such as energy, symmetry, and action. This textbook describes these different approaches to classical mechanics, starting with Newton’s laws before turning to subsequent developments such as the Lagrangian and Hamiltonian approaches. The book emphasises Noether’s profound insights into symmetries and conservation laws, as well as Einstein’s vision of spacetime, encapsulated in the theory of special relativity. Classical mechanics is not the last word on theoretical physics. But it is the foundation for all that follows. The purpose of this book is to provide this foundation.

Much of classical mechanics treats particles as infinitesimally small. But most of our world is not like this. Planets and cats and tennis balls are not infinitesimally small, but have an extended size and this can be important for many applications. The purpose of this chapter is to understand how to describe the complicated motion of extended objects as they tumble and turn.

The purpose of this chapter is to understand how quantum particles react to magnetic fields. There are a number of reasons to do be interested in this. First, quantum particles do extraordinary things when subjected to magnetic fields, including forming exotic states of matter known as quantum Hall fluids. But, in addition, magnetic fields bring a number of new conceptual ideas to the table. Among other things, this is where we first start to see the richness that comes from combining quantum mechanics with the gauge fields of electromagnetism.

The difference between quantum and classical mechanics does not involve just a small tweak. Instead it is a root and branch overhaul of the entire framework. In this chapter we introduce the key concept that underlies this new framework: the quantum state, as manifested in the wavefunction.

Space and time are not what they seem. Their true nature only becomes clear as particles reach the speeds close to the speed of light where some of the common sense ideas start to break down. Indeed, one of major themes of twentieth century physics is that common sense is not a good guide when we look closely at the universe. In this chapter, we start to understand the true nature of space and time, as encapsulated in Einsteins theory of special relativity. We will see many wonderful things, from time slowing down to the lengths shrinking. There will be stories of twins and trains and elementary particles failing to die.

Physicists have a dirty secret: we’re not very good at solving equations. More precisely, humans aren’t very good at solving equations. We know this because we have computers and they’re much better at solving things than we are. This means that we must develop a toolbox of methods so that, when confronted by a problem, we have some options on how to go about understanding whats going on. The purpose of this chapter is to develop this toolbox in the guise of various approximation schemes.