Copyright 2024
Hardback
ISBN 9781032436661
488 Pages 215 Color Illustrations
August 30, 2023 by Chapman & Hall
Differential Geometry and Its Visualization is suitable for graduate level courses in differential geometry, serving both students and teachers. It can also be used as a supplementary reference for research in mathematics and the natural and engineering sciences.
Differential geometry is the study of geometric objects and their properties using the methods of mathematical analysis. The classical theory of curves and surfaces in three-dimensional Euclidean space is presented in the first three chapters. The abstract and modern topics of tensor algebra, Riemannian spaces and tensor analysis are studied in the last two chapters. A great number of illustrating examples, visualizations and genuine figures created by the authorsf own software are included to support the understanding of the presented concepts and results, and to develop an adequate perception of the shapes of geometric objects, their properties and the relations between them.
Extensive, full colour visualisations.
Numerous exercises.
Self-contained and comprehensive treatment of the topic.
1. Curves in Three?dimensional Euclidean Space. 1.1. Points and Vectors. 1.2. Vector?valued Functions of a Real Variable. 1.3. The General Concept of Curves. 1.4. Some Examples of Planar Curves. 1.5. The Arc Length of a Curve. 1.6. The Vectors of the Trihedron of a Curve. 1.7. Frenetfs Formulae. 1.8. The Geometric Significance of Curvature and Torsion. 1.9. Osculating Circles and Spheres. 1.10. Involutes and Evolutes. 1.11. The Fundamental Theorem of Curves. 1.12. Lines of Constant Slope. 1.13. Spherical Images of a Curve. 2. Surfaces in Three?dimensional Euclidean Space. 2.1. Surfaces and Curves on Surfaces. 2.2. The Tangent Planes and Normal Vectors of a Surface. 2.3. The Arc Length, Angles and Gaussfs First Fundamental Coefficients. 2.4. the Curvature of Curves on Surfaces, Geodesic and Normal Curvature. 2.5. The Normal, Principal, Gaussian and Mean Curvature. 2.6. The Shape of a Surface in the Neighbourhood of a Point. 2.7. Dupinfs Indicatrix. 2.8. Lines of Curvature and Asymptotic Lines. 2.9. Triple Orthogonal Systems. 2.10. the Weingarten Equations. 3. The Intrinsic Geometry of Surfaces. 3.1. the Christoffel Symbols. 3.2. Geodesic Lines. 3.3. Geodesic Lines on Surfaces with Orthogonal Parameters. 3.4. Geodesic Lines on Surfaces of Revolution. 3.5. the Minimum Property of Geodesic Lines. 3.6. Orthogonal and Geodesic Parameters. 3.7. Levi?civita Parallelism. 3.8. Theorema Egregium. 3.9. Maps Between Surfaces. 3.10. the Gauss?bonnet Theorem. 3.11. Minimal Surfaces. 4. Tensor Algebra and Riemannian Geometry. 4.1. Differentiable Manifolds. 4.2. Transformation of Bases. 4.3. Linear Functionals and Dual Spaces. 4.4. Tensors of Second Order. 4.5. Symmetric Bilinear Forms and Inner Products. 4.6. Tensors of Arbitary Order. 4.7. Symmetric and Anti?symmetric Tensors. 4.8. Riemann Spaces. 4.9. the Christoffel Symbols. 5. Tensor Analysis. 5.1. Covariant Differentiation. 5.2. the Covariant Derivative of an (R, S)?tensor. 5.3. the Interchange of Order for Covariant Differentiation and Riccifs Identity. 5.4. Bianchifs Identities for the Covariant Derivative of the Tensors of Curvature. 5.5. Beltramifs Differentiators. 5.6. a Geometric Meaning of the Covariant Differentiation, the Levi?civita Parallelism. 5.7. The Fundamental Theorem for Surfaces. 5.8. A Geometric Meaning of the Riemann Tensor of Curvature. 5.9. Spaces With Vanishing Tensor of Curvature. 5.10. An Extension of Frenetfs Formulae. 5.11. Riemann Normal Coordinates and the Curvature of Spaces.
Copyright 2024
Hardback
ISBN 9780367486914
436 Pages 36 B/W Illustrations
August 28, 2023 by Chapman & Hall
The book is devoted to linear and nonlinear ordinary and partial differential equations with constant and variable delay. It considers qualitative features of delay differential equations and formulates typical problem statements. Exact, approximate analytical and numerical methods for solving such equations are described, including the method of steps, methods of integral transformations, method of regular expansion in a small parameter, method of matched asymptotic expansions, iteration-type methods, Adomian decomposition method, collocation method, Galerkin-type projection methods, Euler and Runge-Kutta methods, shooting method, method of lines, finite-difference methods for PDEs, methods of generalized and functional separation of variables, method of functional constraints, method of generating equations, and more.
The presentation of the theoretical material is accompanied by examples of the practical application of methods to obtain the desired solutions. Exact solutions are constructed for many nonlinear delay reaction-diffusion and wave type PDEs that depend on one or more arbitrary functions. A review is given of the most common mathematical models with delay used in population theory, biology, medicine, economics, and other applications.
Delay Ordinary and Partial Differential Equations contains much new material previously unpublished in monographs. It is intended for a broad audience of scientists, university professors, and graduate and postgraduate students specializing in applied and computational mathematics, mathematical physics, mechanics, control theory, biology, medicine, chemical technology, ecology, economics, and other disciplines.
Individual sections of the book and examples are suitable for lecture courses on applied mathematics, mathematical physics, and differential equations, for delivering special courses, and for practical training.
Volume 93 in the series De Gruyter Studies in Mathematics
Maximally subelliptic partial differential equations (PDEs) are a far-reaching generalization of elliptic PDEs. Elliptic PDEs hold a special place: sharp results are known for general linear and even fully nonlinear elliptic PDEs. Over the past half-century, important results for elliptic PDEs have been generalized to maximally subelliptic PDEs. This text presents this theory and generalizes the sharp, interior regularity theory for general linear and fully nonlinear elliptic PDEs to the maximally subelliptic setting.
Discusses sharp regularity theory for linear and fully nonlinear maximally subelliptic PDEs.
Covers Gaussian bounds for the heat equation for maximally subelliptic PDEs.
Presents function spaces adapted to maximally subelliptic PDEs.
Author information
Brian Street, University of Wisconsin-Madison, USA
Analysis
Differential Equations and Dynamical Systems
Geometry and Topology
Mathematics
Volume 95 in the series De Gruyter Studies in Mathematics
The theory of almost periodic functions is a very active field of research for scholars. This research monograph analyzes various classes of multi-dimensional almost periodic type functions with values in complex Banach spaces. We provide many applications of our theoretical results to the abstract Volterra integro-differential inclusions in Banach spaces.
Investigates multi-dimensional $\rho$-almost periodic functions.
Covers almost periodic solutions for various classes of abstract impulsive Volterra integro-differential inclusions.
Author information
long Bio (mandatory for Amazon Top Titles)
Analysis
Differential Equations and Dynamical Systems
Mathematics
The congruences of a lattice form the congruence lattice. Over the last several decades, the study of congruence lattices has established itself as a large and important field with a great number of interesting and deep results, as well as many open problems. Written by one of the leading experts in lattice theory, this text provides a self-contained introduction to congruences of finite lattices and presents the major results of the last 90 years. It features the authorfs signature gProof-by-Pictureh method, which is used to convey the ideas behind formal proofs in a visual, more intuitive manner.
Key features include:
an insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensions
complete proofs, an extensive bibliography and index, and over 180 illustrations
additional chapters covering new results of the last seven years, increasing the size of this edition to 430 pages, 360 statements, and 262 references
This text is appropriate for a one-semester graduate course in lattice theory, and it will also serve as a valuable reference for researchers studying lattices.
g[This] monographcis an exceptional work in lattice theory, like all the contributions by this author. The way this book is written makes it extremely interesting for the specialists in the field but also for the students in lattice theory. ? Cosmin Pelea, Studia Universitatis Babes-Bolyai Mathematica LII (1), 2007
"The book is self-contained, with many detailed proofs presented that can be followed step-by-step. I believe that this book is a much-needed tool for any mathematician wishing a gentle introduction to the field of congruences representations of finite lattices, with emphasis on the more 'geometric' aspects." ? Mathematical Reviews