In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions.
The earlier chapter of this self-contained text provide a route from first principles through standard linear and quadratic algebra to geometric algebra, with Clifford's geometric algebras taking pride of place. In parallel with this is an account, also from first principles, of the elementary theory of topological spaces and of continuous and differentiable maps that leads up to the definitions of smooth manifolds and their tangent spaces and of Lie groups and Lie algebras. The calculus is presented as far as possible in basis free form to emphasize its geometrical flavour and its linear algebra content. In this second edition Dr Porteous has taken the opportunity to add a chapter on triality which extends earlier work on the Spin groups in the chapter on Clifford algebras. The details include a number of important transitive group actions and a description of one of the exceptional Lie groups, the group G2. A number of corrections and improvements have also been made. There are many
In many problems of applied mathematics, science, engineering or economics, an energy expenditure or its analogue can be approximated by upper and lower bounds. This book provides a unified account of the theory required to establish such bounds, by expressing the governing conditions of the problem, and the bounds, in terms of a saddle functional and its gradients. There are several features, including a chapter on the Legendre dual transformation and some of its singularities. Many substantial examples and exercises are included, especially from the mechanics of fluids, elastic and plastic solids and from optimisation theory. The saddle functional viewpoint gives the book a wide scope. The treatment is straightforward, the only prerequisite being a basic knowledge of the calculus of variations. Part of the book is based on final-year undergraduate courses. This is developed into an account which will interest a wide range of students and professionals in applied mathematics, engineer
This treatise deals with modern theory of functional equations in several variables and their applications to mathematics, information theory, and the natural, behavioural and social sciences. The authors have chosen to emphasize applications, though not at the expense of theory, so they have kept the prerequisites to a minimum; the reader need be familiar only with calculus and elementary algebra, and have a basic knowledge of Lebesgue integration. Where, for certain applications, more advanced topics are needed, the authors have included references and explained the results used. Moreover, the book has been designed so that the chapters can be read almost independently of each other, enabling a selection of material to be chosen for introductory and advanced courses. At the end of each chapter are included exercises and further results, some 400 in all, which extend the material presented in the text and also test it. The history of functional equations is well documented in a final
Probabilistic modelling is the most cost-effective means of performance and reliability evaluation of complex dynamic systems. This self-contained text will be welcomed by students and teachers for its no-nonsense treatment of the basic results and examples of their application. The only mathematical background that is assumed is basic calculus. The necessary fundamentals of probability theory are included, as well as an introduction to renewal, Poisson and Markov processes. Models arising in the fields of manufacturing, computing and communications, involving single or multiple service stations and one or more customer classes, are examined in some detail. Both exact and approximate solution methods are discussed, including recent techniques such as spectral expansion. Special attention is devoted to models of systems subject to breakdowns and repairs. Throughout the book, strong emphasis is placed on explaining the ideas behind the results and helping the reader to use them, making t
Probabilistic modelling is the most cost-effective means of performance and reliability evaluation of complex dynamic systems. This self-contained text will be welcomed by students and teachers for its no-nonsense treatment of the basic results and examples of their application. The only mathematical background that is assumed is basic calculus. The necessary fundamentals of probability theory are included, as well as an introduction to renewal, Poisson and Markov processes. Models arising in the fields of manufacturing, computing and communications, involving single or multiple service stations and one or more customer classes, are examined in some detail. Both exact and approximate solution methods are discussed, including recent techniques such as spectral expansion. Special attention is devoted to models of systems subject to breakdowns and repairs. Throughout the book, strong emphasis is placed on explaining the ideas behind the results and helping the reader to use them, making t
Randomness is ubiquitous in nature. Random drivers are generally considered a source of disorder in environmental systems. However, the interaction between noise and nonlinear dynamics may lead to the emergence of a number of ordered behaviors (in time and space) that would not exist in the absence of noise. This counterintuitive effect of randomness may play a crucial role in environmental processes. For example, seemingly 'random' background events in the atmosphere can grow into larger instabilities that have great effects on weather patterns. This book presents the basics of the theory of stochastic calculus and its application to the study of noise-induced phenomena in environmental systems. It will be an invaluable reference text for ecologists, geoscientists and environmental engineers interested in the study of stochastic environmental dynamics.
This book explores the standard problem-solving techniques of multivariable mathematics integrating vector algebra ideas with multivariable calculus and differential equations. Unique coverag
For courses in Precalculus. Providing the rigor of solid mathematics with an engaging and friendly approach As teachers, Ratti and McWaters saw firsthand where their Precalculus and Calculus
This book provides the reader with an elementary introduction to chaos and fractals, suitable for students with a background in elementary algebra, without assuming prior coursework in calculus or phy
For briefer traditional courses in elementary differential equations that science, engineering, and mathematics students take following calculus. The Sixth Edition of this widely adopted b
Calculus and Its Applications, Eleventh Edition, remains a best-selling text because of its accessible presentation that anticipates student needs. The writing style is ideal for today’s students, pro
Since the 1970s social scientists and scientists in a variety of fields - psychology, sociology, education, psychiatry, economics and engineering - have been interested in problems that require the statistical analysis of data over time and there has been in effect a conceptual revolution in ways of thinking about pattern and regularity. This book is a comprehensive introduction to all the major time-series techniques, both time-domain and frequency-domain. It includes work on linear models that simplify the solution of univariate and multivariate problems. The author begins with a non-mathematical overview: throughout, he provides easy-to-understand, fully worked examples drawn from real studies in psychology and sociology. Other, less comprehensive, books on time-series analysis require calculus: this presupposes only a standard introductory statistics course covering analysis of variance and regression. The chapters are short, designed to build concepts (and the reader's confidence)
Continuous-Time Models in Corporate Finance synthesizes four decades of research to show how stochastic calculus can be used in corporate finance. Combining mathematical rigor with economic intuition,
Presents an entirely new approach to introductory physics within a calculus-based conceptual and a mathematical framework. It offers an approach to presenting the material that is more gradual than ex
In studying the dynamics of populations, whether of animals, plants or cells, it is crucial to allow for intrinsic delays, due to such things as gestation, maturation or transport. This book is concerned with one of the fundamental questions in the analysis of the effect of delays, namely determining whether they effect the stability of steady states. The analysis is presented for one or two such delays treated both as discrete, where an event which occurred at a precise time in the past has an effect now, and distributed, where the delay is averaged over the population's history. Both of these types occur in biological contexts. The method used to tackle these questions is linear stability analysis which leads to an understanding of the local stability. By avoiding global questions, the author has kept the mathematical prerequisites to a minimum, essentially advanced calculus and ordinary differential equations.
Here is a textbook of intuitive calculus. The material is presented in a concrete setting with many examples and problems chosen from the social, physical, behavioural and life sciences. Chapters include core material and more advanced optional sections. The book begins with a review of algebra and graphing.
For one-semester courses in Finite Math & Applied Calculus or Mathematics for Business. Built-in guidance that helps students “get the idea.” College Mathematics for Business, Economics, Lif
Stories abound about the lengths to which middle- and upper-middle-class parents will go to ensure a spot for their child at a prestigious university. From the Suzuki method to calculus-based physics,
This book provides a solid introduction to the classical and statistical theories of thermodynamics while assuming no background beyond general physics and advanced calculus. Though an acquaintance w
