Basic electrical quantities

Matter is made of atoms, which can be conveniently visualised as a small, positively charged nucleus encircled by negatively charged electrons (the naming of charges as positive and negative was arbitrary but is long established). Usually, the amounts of positive and negative charges are equal, so they balance to give electrical neutrality; only when there is imbalance does a body have a charge and its electrical properties become apparent.

Introduction

Unrecognised cavities beneath a site could lead to settling or collapse of the structure, or, in the case of dams and settling ponds, allow the contents to escape. Cavities may be natural or artificial.

Basic ideas in geothermics

Introduction

Temperature increases with depth in the Earth; this is why, for instance, deep mines are hot and deeply buried rocks are thermally metamorphosed. The increase is partly because heat is coming out of the hot interior, just as heat leaks out of a hot oven, and partly because heat is being produced within the rocks by radioactive heating.

Radioactive radiations

The previous chapter explained how radioactivity could be used to date rocks because isotopes decay from one element to another. This chapter is mainly concerned with the ‘radiations’ that accompany the decays, as a way of detecting and identifying the source elements. There are three principal types of radiation, all of which originate from the nuclei of radioactive atoms (Section 15.12.1). α-particles consist of two protons and two neutrons, and so have a positive electrical charge (ultimately, they each combine with two electrons to form helium atoms, and this is the origin of the helium used to inflate balloons). β-particles are electrons – produced when a proton converts to a neutron plus a electron – and so have a negative charge. Because of their electric charges, α-and β-particles cannot travel far through matter, no more than a few centimetres in air, or a few millimetres of rock, and so are little used in surveying.

Introduction

In Part I, deciding which method to use in any of the examples given was not a problem, for they were chosen to illustrate the particular method being described, but when a geological problem is first encountered it is necessary to decide which – if any – geophysical methods to use and how best to employ them. Choosing the most suitable one or combination needs experience and perhaps some luck, but considering the following questions should narrow the choice.

Does the problem have geophysical expression?

Geophysical surveys do not respond to geological features as such, but to differences in physical properties, so the first requirement is that the geological situation has geophysical expression; that is, there must be some related subsurface body or structure that can be detected geophysically. For example, a granite pluton, which rose into place because of its low density, gives rise to a negative gravity anomaly (Fig. 8.16), and this may be used to locate it and estimate its size. In this example, the geophysical expression – the negative anomaly – is directly due to the body to be detected because its density is an intrinsic property of the granite, but sometimes geophysical expression is indirect. For example, a fault may be detectable by a seismic reflection survey if it has produced a vertical offset in subhorizontal layers (Fig. 7.10) but not if there are no layers or they are not offset vertically; or a concealed shaft may be directly detected by its negative gravity anomaly, but indirectly, for example, by a magnetic survey if it happens to contain ferrous objects (Section 27.2.3).

The purpose of Modern Mathematical Methods for Physicists and Engineers is to help graduate and advanced undergraduate students of the physical sciences and engineering acquire a sufficient mathematical background to make intelligent use of modern computational and analytical methods. This book responds to my students’ repeated requests for a mathematical methods text with a modern point of view and choice of topics.

For the past fifteen years I have taught graduate courses in computational and mathematical physics. Before introducing the course on which this book is based, I found it necessary, in courses ranging from numerical methods to the applications of group theory in physics, to summarize the rudiments of linear algebra and functional analysis before proceeding to the ostensible subjects of the course. The questions of the students who studied early drafts of this work have helped to shape the presentation. Some students working concurrently in nearby telecommunication, semiconductor, or aerospace, industries have contributed significantly to the substance of portions of the book.

The following is an example of the situations that motivated me to take the time to write a mathematical methods text that breaks significantly with the past: Every semester, students come to my office, puzzled over numerical modeis in which minor changes in the data produce drastic changes in the Outputs. Unfortunately most of these students lack the mathematical background needed to conceptualize some of the most common problems of numerical computation. For an engineer, and for the increasingly large fraction of physics graduates who make careers in numerical modeling or electrical engineering, conceptual understanding of analytical and numerical modeis is an absolutely essential ingredient of successful designs. A Computer can be a tool for understanding, and not merely a means for obtaining a numerical answer of unknown reliability and significance, only in the hands of those who understand the foundations and potential shortcomings of numerical methods. Yet the traditional mathematical methods taught to students in engineering and physics for most of the twentieth Century do not provide a sufficient background even for introductory graduate texts on many important contemporary topics, of which numerical computation is only one.

What upper-level undergraduate and first-year graduate students in physics and engineering tend consistently to lack is an understanding of basic mathematical structures - groups, rings, fields, and vector Spaces - and of mappings that preserve these structures.