Book Description

Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes.It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. It overlaps with convex geometry, and employs many tools from that area including integral geometry.It also has connections to geometric probing in robotics and to stereology.The main text contains a rigorous treatment of the subject starting from basic concepts and moving up to the research frontier: seventy-two unsolved problems are stated. Each chapter ends with extensive notes, historical remarks, and some biographies. This comprehensive work will be invaluable to specialists in geometry and tomography; the opening chapters can also be read by advanced undergraduate students. ... Read more

Reviews (1)

Introduction to and complete survey of Geometric Tomography
My MSc research was in the subject of Geometric Tomography (more specifically, parallel X-rays of connected sets), and Richard Gardner's book was an invaluable reference book to me during my research, especially chapter 1 (Parallel X-rays of planar convex bodies) and 2 (Parallel X-raysin n dimensions).

But this book covers much more - everything that couldbe considered "Geometric Tomography". It assumes no priorknowledge of the subject, and then builds the entire theory of GeometricTomography (to which Gardner has been one of the most importantcontributers), step by step, building on and referring to almost 550journal articles and books.

Gardner's "Geometric Tomography" isan invaluable tool to anybody who is seriously interested in the subject ofGeometric Tomography, and and must-have to any serious Mathematics library. ... Read more

Book Description

Reviews (4)

Took the class and the book
I had the class from Prof. Parker ~20 years ago. (BS Mathematics 83 from SIU)It was a wonderful class and this is a wonderful book.I still have my signed! copy.I am now a professor of EE and a large research university and this is still a subject that I love.Credit that to the Book and Prof. Parker.

A solid introduction
It is hard to disagree with the idea that one must pursue the learning of mathematics in way that might be at odds with its axiomatic structure. One can pursue the study of differentiable manifolds without ever looking at a book on classical differential geometry, but it is doubtful that one could appreciate the underlying ideas if such a strategy were taken. Some background in linear algebra, topology, and vector calculus would allow one to understand the abstract definition of a differentiable manifold. However, to push forward the frontiers of the subject, or to apply it, one must have a solid understanding of its underlying intuition.

Thus a study of classical differential geometry is warranted for someone who wants to do original research in the area as well as use it in applications, which are very extensive. Differential geometry is pervasive in physics and engineering, and has made its presence known in areas such as computer graphics and robotics. In this regard, the authors of this book have given students a fine book, and they emphasize right at the beginning that an undergraduate introduction to differential geometry is necessary in today's curriculum, and that such a course can be given for students with a background in calculus and linear algebra. They also do not hesitate to use diagrams, without sacrificing mathematical rigour. Too often books in differential geometry omit the use of diagrams, holding to the opinion that to do so would be a detriment to mathematical rigour. Much is to be gained by the reading and studying of this book, and after finishing it one will be on the right track to begin a study of modern differential geometry.

A Perfect Introduction
A Must !!!After reviewing a few dozen books in the subject, this is without any doubt one of the best. It it written with rare clarity, and givesenough motivation and examples to understand the more abstract and difficult aspects of the field. The book is intended for advanced undergraduate (with good understanding of linear algebra and calculus III) and should be read prior to an abstract course in differential geometry (such as is covered in the books of Warner and Hicks).

Another Differential Geometry Book - So So
This book I also purchased as a resource for studying differential geometry. It's a little bit better than the one by Thorpe, but not by much. The text is dedicated to the 'hard-core' mathematical, and even they would have to have some experience/guidance in this subject.I'm a self-learning type of guy, with an MS in physics. Too many questions arise to justify this book for the self-learner. There are problems, and a FEW examples. ... Read more

Book Description

This introduction treats the classical isoperimetric inequality in Euclidean space and contrasting rough inequalities in noncompact Riemannian manifolds. In Euclidean space the emphasis is on a most general form of the inequality sufficiently precise to characterize the case of equality, and in Riemannian manifolds the emphasis is on those qualitiative features of the inequality that provide insight into the coarse geometry at infinity of Riemannian manifolds. The treatment in Euclidean space features a number of proofs of the classical inequality in increasing generality, providing in the process a transition from the methods of classical differential geometry to those of modern geometric measure theory; and the treatment in Riemannian manifolds features discretization techniques, and applications to upper bounds of large time heat diffusion in Riemannian manifolds. The result is an introduction to the rich tapestry of ideas and techniques of isoperimetric inequalities, a subject that has its beginnings in classical antiquity and which continues to inspire fresh ideas in geometry and analysis to this very day--and beyond! ... Read more

Book Description

Finite element, finite volume and finite difference methods use grids to solve the numerous differential equations that arise in the modelling of physical systems in engineering.Structured grid generation forms an integral part of the solution of these procedures.Basic Structured Grid Generation provides the necessary mathematical foundation required for the successful generation of boundary-conforming grids and will be an important resource for postgraduate and practising engineers.

The treatment of structured grid generation starts with basic geometry and tensor analysis before moving on to identify the variety of approaches that can be employed in the generation of structured grids.The book then introduces unstructured grid generation by explaining the basics of Delaunay triangulation and advancing front techniques.A companion website fully supports this book by providing numerical codes in FORTRAN 77/90 for both structured and unstructured grid generation which will help the reader to develop their understanding and make progress in grid generation.

* A practical, straightforward approach to this complex subject for engineers and students.
* A key technique for modelling physical systems.
* Companion website provides free access to grid generation codes. ... Read more

Book Description

The Radon transform is an important topic in integral geometry which deals with the problem of expressing a function on a manifold in terms of its integrals over certain submanifolds. Solutions to such problems have a wide range of applications, namely to partial differential equations, group representationsm X-ray technology, nuclear magnetic resonance scanning, and tomography.This second edition, significantly expanded and updated, presents new material taking into account some of the progress in the field since 1980. ... Read more

Reviews (1)

Theory
I saw a previous (1st) edition of this book in my college library.It's too theoretical for engineers--useless for engineers.It's written for mathematicians. I thought this would help since u can't see what's inside online. ... Read more

Book Description

The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text covers symplectomorphisms, local forms, contact manifold, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moments maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. ... Read more

Reviews (1)

an engineer's view
Each chapter in the book is by different authors and so the level of detail, notation, etc. is uneven. The chapters are more than just reprints of previous materials and are self-contained. The early foundation chapters are written from a mathematician's perspective, which is wholly unsatisfying for an engineer. There is a minor emphasis on binary images throughout the book, but binary images are not treated in any depth. In general, the algorithm and application chapters are more overview than details. Examples are almost nonexistent which means this would be an extremely poor textbook even for upper-level classes. The example in chapter 11 is full of errors and a bit incomplete. Overall, the chapters are much better than any one of the conference papers they are based on, but you will still need to consult the references to really use any of the information. ... Read more

Book Description

Smooth Manifolds and Observables is about the differential calculus, smooth manifolds, and commutative algebra. While these theories arose at different times and under completely different circumstances, this book demonstrates how they constitute a unified whole. The motivation behind this synthesis is the mathematical formalization of the process of observation in classical physics.

The main objective of this book is to explain how differential calculus is a natural part of commutative algebra. This is achieved by studying the corresponding algebras of smooth functions that result in a general construction of the differential calculus on various categories of modules over the given commutative algebra. It is shown in detail that the ordinary differential calculus and differential geometry on smooth manifolds turns out to be precisely the particular case that corresponds to the category of geometric modules over smooth algebras. This approach opens the way to numerous applications, ranging from delicate questions of algebraic geometry to the theory of elementary particles.

This unique textbook contains a large number of exercises and is intended for advanced undergraduates, graduate students, and research mathematicians and physicists. ... Read more

Book Description

Designed for advanced undergraduate or beginning graduate study, this text contains an elementary introduction to continuous groups and differential invariants; an extensive treatment of groups of motions in euclidean, affine, and riemannian geometry; and development of the method of integral formulas for global differential geometry.
... Read more

Reviews (2)

Not only for pure mathematician
I find the book very interesting: it's a very good presentation of "classical problems with modern methods" in Differential Geometry. It's appreciable for the selection of topics and their logical order, the clarity of their exposition (based on the use of modern terminology), the set of proposed problems and the relative results and the list of references at the end of each chapter.

if interested in this topic
Guggenheimer gives a good introduction to the topic, although a good background in mathematics (analysis) is very helpful. While not one of the easiest books to read, it serves as a good supplemental text to anyone browsing through other texts on differential geometry. ... Read more

Book Description

This book provides a comprehensive look at the Schwarz-Christoffel transformation, including its history and foundations, practical computation, common and less common variations, and many applications in fields such as electromagnetism, fluid flow, design and inverse problems, and the solution of linear systems of equations. It is an accessible resource for engineers, scientists, and applied mathematicians who seek more experience with theoretical or computational conformal mapping techniques. The most important theoretical results are stated and proved, but the emphasis throughout remains on concrete understanding and implementation.There is a brief appendix illustrating the use of the Schwarz-Christoffel Toolbox for MATLAB, the state-of-the-art package for computation of these maps. ... Read more

Reviews (1)

Excellent book
This short book is a good place to start learning about the theory of Reidemeister torsions. Masterfully written by one of the leading experts on the subject, it follows the standard of exposition set by Milnor in his famous survey on this subject 40 years ago. The book is very easy to read, the necessary background is covered economically, but in details. You can find here many results that received the first treatment in book form and some classical results with illuminating explainations. The last chapter briefly indicates recent developments in connection with Seiberg-Witten invariants of 3-manifolds.

The book is mainly an introduction and concerns mostly with combinatorial manifolds and knot theory. As the author stated, many other aspects of the theory (e.g. the analytical theory) are left out. ... Read more

Book Description

Elliptic equations of critical Sobolev growth have been the target of investigation for decades because they have proved to be of great importance in analysis, geometry, and physics. The equations studied here are of the well-known Yamabe type. They involve Schrödinger operators on the left hand side and a critical nonlinearity on the right hand side.

A significant development in the study of such equations occurred in the 1980s. It was discovered that the sequence splits into a solution of the limit equation--a finite sum of bubbles--and a rest that converges strongly to zero in the Sobolev space consisting of square integrable functions whose gradient is also square integrable. This splitting is known as the integral theory for blow-up. In this book, the authors develop the pointwise theory for blow-up. They introduce new ideas and methods that lead to sharp pointwise estimates. These estimates have important applications when dealing with sharp constant problems (a case where the energy is minimal) and compactness results (a case where the energy is arbitrarily large). The authors carefully and thoroughly describe pointwise behavior when the energy is arbitrary.

Intended to be as self-contained as possible, this accessible book will interest graduate students and researchers in a range of mathematical fields.

Book Description

For the past 25 years, the Geometrization Program of Thurston has been a driving force for research in 3-manifold topology.This has inspired a surge of activity investigating hyperbolic 3-manifolds (and Kleinian groups), as these manifolds form the largest and least well-understood class of compact 3-manifolds.Familiar and new tools from diverse areas of mathematics have been utilized in these investigations, from topology, geometry, analysis, group theory, and from the point of view of this book, algebra and number theory.This book is aimed at readers already familiar with the basics of hyperbolic 3-manifolds or Kleinian groups, and it is intended to introduce them to the interesting connections with number theory and the tools that will be required to pursue them.While there are a number of texts which cover the topological, geometric and analytical aspects of hyperbolic 3-manifolds, this book is unique in that it deals exclusively with the arithmetic aspects, which are not covered in other texts. Colin Maclachlan is a Reader in the Department of Mathematical Sciences at the University of Aberdeen in Scotland where he has served since 1968. He is a former President of the Edinburgh Mathematical Society. Alan Reid is a Professor in the Department of Mathematics at The University of Texas at Austin. He is a former Royal Society University Research Fellow, Alfred P. Sloan Fellow and winner of the Sir Edmund Whittaker Prize from The Edinburgh Mathematical Society. Both authors have published extensively in the general area of discrete groups, hyperbolic manifolds and low-dimensional topology. ... Read more

Book Description

This series of International Conferences on Difference Equations and Applications has established a tradition within the mathematical community bringing together scientists from many different areas of research to highlight current interests, challenges and unresolved problems. This volume comprises selected papers presented at the Fifth International Conference on Difference Equations, which was held at Temuco, Chile and Provides a Valuable source of reference for graduates and researchers. ... Read more

Book Description

Devoted to the classical analytical mechanics of systems with a finite number of degrees of freedom, with special attention given to some nonstandard problems, both theoretical and practical. Presents the geometric formulation of analytical mechanics in terms of tangent and cotangent bundles and symplectic and contact manifolds. In contrast to purely formal treatments, the author justifies in physical terms the symplectic structure presupposed by classical Hamiltonian mechanics. The result is that the well-known structures of the Hamilton-Jacobi theory are given a deep geometrical interpretation. ... Read more

Reviews (1)

You have got to be kidding.
Actually, I have not read the book, and considering the price, I would be philosophically opposed to doing so. ... Read more