One of the prominent advantages of the gauge formalism as a field theory is its sophisticated mathematical structure, being based on analytical mechanics. Everything about the system dynamics that we need can be derived by rote out of a Lagrangian density of the system, which itself can be determined uniquely based on the prescribed symmetry underlying in the physical phenomenon we want to describe. In our case, we can find how the dislocation and defect fields should be incorporated into the continuum theory of elasticity, with direct correspondences to the differential geometrical (DG) counterparts introduced in Chapter 6. Also the formalism can provide us with a bridge between the DG pictures and the method of quantum field theory (QFT) discussed in Chapter 8 via the Lagrangian density.

Flexure is an important feature of deep-sea trench–outer rise systems. In particular, the topographic rises seaward of deep-sea trenches can be explained by vertical and horizontal loads that act on the approaching oceanic lithosphere. While the source of these loads is not always clear, they combine to flex the lithosphere downwards by a few kilometres at the trench and upwards by several hundreds of metres in the region of the topographic rise. Many of the differences in the bathymetry of deep-sea trench–outer rise systems can be explained by variations in Te of the approaching plate. Others may be due to variations in the state of stress that exist across the plate boundary. There is evidence that the flexure is large enough at some trenches to locally weaken the lithosphere and cause it to fail through normal faulting. Seamounts and oceanic islands that ‘ride’ the topographic rise will experience uplift and then subsidence as they are carried by plate motions into the trench. This will also be the case for continents that arrive at a trench because of the subduction of oceanic crust. Uplift is in the process of affecting New Caledonia and other neighbouring islands, and flexure seaward of trenches may explain several erosional unconformities that are observed in the ancient rock record.

This chapter introduces some of the basic tools of a cosmologist, including scale factor, redshift, and comoving distance. We start with the Hubble law, which is a key consequence of the expanding universe. Next, we cover the possible geometries of space (positively and negatively curved, and flat), and the associated Friedmann--Lemaître--Robertson--Walker metric that describes them. This leads us to define distance measures in cosmology, and introduce the Friedmann equation that describes the evolution of the universe given its contents. We end by discussing the role of critical density and curvature.

One of the problems with the concept of spacetime is that it is hard for us to actually appreciate the implications of living in a curved spacetime, and the origin of this difficulty is that our local spacetime is essentially flat! Hence, all of our understanding of physics – of 'how things work' – has been built on the basis of perceptions that take place in almost flat spacetime. This chapter will provide a pragmatic approach to the measurement of spacetime by illustrating how it is actually not too difficult to obtain an estimate of local curvature by using simple physical quantities, such as the mass and the size of the object. In this manner, we will be able to appreciate that the curvature on Earth is only a few parts in a billion, hence explaining why we perceive everything in the actual absence of curvature. we will learn how to actually bend spacetime reaching the extreme values that are encountered near a neutron star and a black hole, both of which will be discussed more in detail in the following chapters.

The end result of Einstein’s revolutionary vision is that gravity is simply the manifestation of the curvature of spacetime. This is a concept that has a deep significance and is at the heart of the Einstein field equations. This Chapter will explain why we need to introduce the idea of 'spacetime' and how we can define the concept of spacetime curvature in this description. Starting from the example of a spacetime empty of matter – that is, a flat spacetime – we will move to the example of a spacetime containing matter and energy – that is, a curved spacetime. This chapter will explain why we find the description of gravity proposed by Newton very reasonable and why we have trouble appreciating the new vision proposed by Einstein. We will contrast the two descriptions with a simple example and show how the very same physical phenomenon – the orbit of the Earth around the Sun – can be seen with very different explanations by Newton and Einstein.

We equip differentiable manifolds with a metric and introduce differential geometry, which provides the mathematical formalism underlying the theory of general relativity and many other applications in different areas of physics, science, and engineering.

The objective of this chapter is to review the Taylor series expansion and discuss its usage in error estimation. The unique value of Taylor series expansion is often neglected. The major assumption is that a function must be infinitely differentiable to use the Taylor series expansion. In real applications in oceanography, however, hardly there is a need to worry about a derivative higher than the 3rd order, although one may think of some exceptions. The point is, there is rarely a need in oceanography and other environmental sciences to actually consider calculating a very high order derivative, unless for theoretical investigations or under special situations. So the application of Taylor series expansion usually only involves the first two derivatives. In this chapter, some simple examples are included for a better understanding of the applications.

As an appendix, we can look briefly at the central ideas of General Relativity (though we are limited, since much of the maths is beyond our scope). We prepare the ground with a number of thought experiments, and then discuss, in outline, the geometrical ideas we have to use. We can get a sense of what Einstein's equation is doing, and we look at some solutions of Einstein's equation (including the Schwarzschild metric), describing possible spacetimes.

The first chapter contains a räsumä of the cosmology treating the homogeneous and isotropic universe. The Friedmann equations are derived and the thermal history of the Universe is discussed in some detail. Special emphasis is laid on the process of recombination and the decoupling of photons from the cosmic uid. Nucleosynthesis and cosmic in ation are also discussed.

Rigid origami describes origami where each face of the crease pattern is flat, as if made from stiff metal.Modeling rigid origami with matrices allows one to describe materials that have been folded into a three-dimensional shape, as opposed to flat origami.This chapter describes this matrix model and proves its key features.In addition, a generalization of Maekawa’s Theorem for three-dimensional rigid origami is introduced, as is modeling rigid origami with the Gauss map from differential geometry.The latter turns out to be a useful tool for the remainder of the book.

Presents relevant aspects of topology, such as homeomorphism, fiber and vector bundles, connection and curvature, parallel transport, and holonomy, and ends with establishing the relevance of topology to physics.

Describes Berry’s phase, connection, and curvature, and derives the Chern topological number. It presents two pedagogically important but distinct examples: a two-level system, with its concomitant “magnetic monopole," and the molecular Aharonov–Bohm effect, where the interplay between the quantum electronic andionic motions leads to fascinating topological manifestations.