In the series De Gruyter Textbook
This textbook offers a different approach to classical textbooks in Differential Geometry. It includes practical examples and over 300 advanced problems designed for graduate students in various fields, such as fluid mechanics, gravitational fields, nuclear physics, electromagnetism, solid-state physics, and thermodynamics. Additionally, it contains problems tailored for students specializing in chemical, civil, and electrical engineering and electronics. The book provides fully detailed solutions to each problem and includes many illustrations to help visualize theoretical concepts.
The book introduces Frenet equations for plane and space curves, presents the basic theory of surfaces, and introduces differentiable maps and differentials on the surface. It also provides the first and second fundamental forms of surfaces, minimal surfaces, and geodesics. Furthermore, it contains a detailed analysis of covariant derivatives and manifolds.
The book covers many classical results, such as the Lancret Theorem, Shell Theorem, Joachimsthal Theorem, and Meusnier Theorem, as well as the fundamental theorems of plane curves, space curves, surfaces, and manifolds.
A different and more accessible format of existing textbooks in Differential Geometry.
Contains many engineering problems and illustrations.
Propose alternative ways of proving classical theorems and a novel exposition of well-known topics.
Frontmatter
Publicly Available I
Preface
Publicly Available V
Contents
Publicly Available VII
1 Curves in
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2 Plane curves
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3 General theory of surfaces
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4 Fundamental equations of a surface. Special classes of surfaces
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5 Differential forms
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6 The nature connection
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7 Riemannian manifolds
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Index
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This book is a self-contained introduction to the theory of Brownian motions and heat kernels on matrix Lie groups and manifolds, with an emphasis on the study of area type functionals. It offers graduate students a systematic account of the subject and serves as a convenient resource and reference for more experienced mathematicians. The book emphasizes methods rather than results and takes the reader to the frontiers of current research, starting with carefully motivated examples and constructions. These aspects are supported by the inclusion of several bibliographic notes at the end of each chapter and appendices at the end of the book.
This book can be used as a self-study guide for readers interested in the interplay between geometry and probability or as a textbook for a special topics course.
Polar spaces are the natural geometries for the classical groups. Due to
the stunning simplicity of an axiom system found by Buekenhout and Shult,
they play a central role in incidence geometry and also appear as combinatorial
objects in many disciplines such as discrete mathematics, graph theory,
finite geometry and coding theory. They can also be viewed as a class of
spherical Tits buildings and as such were classified by Jacques Tits using
pseudo-quadratic forms and octonion algebras. Polar spaces bridge the areas
of group theory, algebra, combinatorics and incidence geometry.
These lecture notes arose from a master course in Ghent, Belgium, taught annually between 2010 and 2024. Besides many basic and general geometric properties of polar spaces, it contains a complete algebraic description of all polar spaces of rank at least 3, linking them with polarities in projective spaces. The discussion of the related classical groups is limited to the study of axial and central elations. The classification of top-thin polar spaces is included in detail. Triality in top-thin polar spaces of rank 4 is explained both geometrically and algebraically. The last chapter introduces parapolar spaces, which are geometric structures using polar spaces as building blocks. This opens the door for exploring geometries related to the exceptional groups. An appendix explains composition algebras, which are used to describe the so-called non-embeddable polar spaces and triality.
Front matter
Download pp. v
Preface
Download p. vii
Introduction
Download pp. ixiv
Contents
Download pp. xvvii
1 Definition and basic properties pp.・9
2 Shult spaces and the one-or-all axiom pp. 21・1
3 Generalised polarities and reflexive forms pp. 33・5
4 Polar spaces from pseudo-quadratic forms pp. 57・3
5 Non-embeddable polar spaces pp. 75・6
6 Diagrams and oriflamme geometries pp. 87・00
7 Central and axial collineations pp. 101・09
8 The geometric principle of triality pp. 111・24
9 Parapolar spaces pp. 125・43
A Cayleyickson division algebras and Moufang planes pp. 14559
References p. 161
Index pp. 16364
Differential-algebraic equations are a widely accepted tool for the modeling and simulation of constrained dynamical systems in numerous applications, such as mechanical multibody systems, electrical circuit simulation, chemical engineering, control theory, fluid dynamics and many other areas.
In the second edition of this textbook a systematic and detailed analysis
of initial and boundary value problems for differential-algebraic equations
is provided. The analysis is developed from the theory of linear constant
coefficient systems via linear variable coefficient systems to general
nonlinear systems. Further sections on control problems, optimal control,
stability theory, generalized inverses of differential-algebraic operators,
generalized solutions, differential equations on manifolds, and differential-algebraic
equations with symmetries complement the theoretical treatment of initial
value problems. Two major classes of numerical methods for differential-algebraic
equations (Rungeutta and BDF methods) are discussed and analyzed with respect
to convergence and order. A chapter is devoted to index reduction methods
that allow the numerical treatment of general differential-algebraic equations.
The analysis and numerical solution of boundary value problems for differential-algebraic
equations is presented, including multiple shooting and collocation methods.
A chapter on further selected topics dealing with overdetermined consistent
systems, root finding, pathfollowing, hybrid systems, and dissipative Hamiltonian
systems completes the book.
A prerequisite for the reader is the standard course on the theory and numerical solution of ordinary differential equations. Numerous examples and exercises make the book suitable as a course textbook or for self-study.
For the first edition of this book, please click here.
Frontmatter
Download pp. iv
Preface of first edition
Download p. v
Preface of second edition
Download p. vii
Contents
Download pp. ixi
1 Introduction Download pp. 32
2 Linear differential-algebraic equations with constant coefficients pp. 137
3 Linear differential-algebraic equations with variable coefficients pp. 8934
4 Non-linear differential-algebraic equations pp. 23509
5 Numerical methods for strangeness-free problems pp. 31382
6 Numerical methods for index reduction pp. 38310
7 Boundary value problems pp. 41169
8 Further selected topics pp. 47197
Final remarks p. 499
References pp. 50120
Index pp. 52126