Copyright 2026
ISBN 9781032959788
220 Pages 14 B/W Illustrations
September 2, 2025 by Chapman & Hall
This text is designed to update the Differential Geometry course by making it more relevant to today’s students. This new approach emphasizes applications and computer programs aimed at twenty-first-century audiences. It is intended for mathematics students, applied scientists, and engineers who attempt to integrate differential geometry techniques in their work or research.
The course can require students to carry out a daunting amount of time-consuming hand computations like the computation of the Christoffel Symbols. As a result, the scope of the applied topics and examples possible to cover might be limited. In addition, most books on this topic have only a scant number of applications.
The book is meant to evolve the course by including topics that are relevant to students. To achieve this goal the book uses numerical, symbolic computations, and graphical tools as an integral part of the topics presented. The provides students with a set of Maple/Matlab programs that will enable them to explore the course topics visually and in depth. These programs facilitate topic and application integration and provide the student with visual enforcement of the concepts, examples, and exercises of varying complexity.
This unique text will empower students and users to explore in-depth and visualize the topics covered, while these programs can be easily modified for other applications or other packages of numerical/symbolic languages. The programs are available to download to instructors and students using the book for coursework.
Chapter 1. Curves and Transformations in Three Dimensions
1.1 Curves and Parameterizations
1.2 Properties of Curves in R3
1.3 Curvature and Torsion
1.4 Helices
1.5 Plane Curves
1.6 Quaternions and Rotations in Three Dimensions
Appendix A The Catenary
Appendix B Astrobiology and Curves in R3
Appendix C Maple
Chapter 2. Introduction to Classical Riemannian Geometry
2.1 Surfaces in R3
2.2 Tangent Planes
2.3 First Fundamental Form of a Surface
2.4 Other Fundamental Forms
2.5 Manifolds in Rm
2.6 Tensors
Appendix A Maple and Matlab Programs
Chapter 3. Tensor Analysis on Riemann Manifolds
3.1 Geodesics
3.2 Examples of Geodesics
3.3 Covariant Differentiation
3.4 Riemann and Ricci Tensors
35 Parallel Transportation of Vectors
3.6Applications to General Relativity
3.7The Torus in R3
Appendix A Maple and MatLab programs
Chapter 4. Basic Topology and Analysis
4.1 Basic Notions of Topology
4.2 Basic notions from Analysis
4.3 Summary
Chapter 5. Differential Manifolds
5.1 Introduction
5.2 Charts and Atlases
5.3 Orientation
5.4 Differentiable Mappings
5.5 The Riemann Sphere-Stereographic mappings
5.6 Stereographic Atlas
5.7 Grassmann Manifolds
Chapter 6. Differentiation on Manifolds
6.1 Differentiation
6.2 Tangent Vectors
6.3 Derived Mappings
Chapter 7. Vectors and Bundles
7.1 The Tangent Bundle
7.2 Cotangent Bundle
7.3 Brief Linear Algebra Review
7.4 Cotagent Bundle on a Manifold
7.5 Vector Fields
Chapter 8. Differential forms
8.1 Exterior Products
8.2 Differential forms
8.3 Hodge Star Operator
8.4 Inverse of Poincare Lemma
8.5 Applications
8.6 Differential Geometry of Manifolds in R3
8.7 Frames in R3
8.8 Smooth Manifolds in R3
8.9 The Laplace Operator
8.10 Maxwell Equations in Free Space Appendix A Functionals over a Vector Space
Chapter 9. Integration on Manifolds in R3
9.1 Integration in one-dimensions
9.2 Volumes in R3
Chapter 10. Integration on Manifolds
10.1 Simplicies in Euclidean Space
10.2 Simplicies and Chains on Manifolds
10.3 Integration of Forms on Manifolds
10.4 Integral Theorems for Surfaces in R3
10.5 Relative Tensors and Integration on Non-Orientable Manifolds
Chapter 11. Symmetry and Lie Groups
11.1 Definition of a Group
11.2 Introduction to Symmetry
11.3 Lie Groups: (“operational definition”)
11.4 Spaces with Indefinite Metrics
11.5 Lie Algebras
11.6 Manifolds and Lie Groups
11.7 The Exponential Map
11.8 Lie transformation Groups
11.9 Applications
Copyright 2026
Hardback
ISBN 9780367774691
272 Pages 6 B/W Illustrations
September 24, 2025 by Chapman & Hall
Introduction to Special Functions for Applied Mathematics introduces readers to the topic of special functions, with a particular focus on applications. Designed to build swiftly from the more basic special functions towards more advanced material, the book is ideally suited for an intensive one semester course. Complemented with various solved examples and exercises to support students and instructors, the book can be used for both self-study and directed learning.
. Suitable for graduate level students or beginning PhD students in mathematics, physics, statistics, and economics
. No previous background in complex analysis required
. Numerous solved examples and exercises
Preface 1 Preliminaries 2 Gamma and Beta functions 3 Digamma, Trigamma and Polygamma 4 Digamma, Trigamma and Polygamma 5 A synopsis on second order linear ODE 6 Airy and Hermite functions 7 Bessel functions: part 1 8 Bessel functions: part 2 9 Gauss hypergeometric function 10 Generalized Hypergeometric functions of one variable 11 Applications and exercises Bibliography Analytic Index
Copyright 2026
ISBN 9781041048343
300 Pages 8 Color & 63 B/W Illustrations
September 2, 2025 by Chapman & Hall
The book contains hundreds of engaging, class-tested statistics exercises (and detailed solutions) that test students’ understanding of the material. Many are educational in their own right?for example, baseball managers who played professional ball were often catchers; stocks that are deleted from the Dow Jones Industrial Average have generally done better than the stocks that replaced them; athletes may not get hot hands but they often get warm hands with modest improvements in their success probabilities.
Introduction 1. Mean, Median, Descriptive Statistics 2. Graphs Good, Bad, and Ugly 3. Misleading Data 4. Probabilities 5. Bayes’ Rule 6. Monty Hall Problems 7. Binomial Distribution 8. Law of Averages 9. Normal Distribution 10. One-Sample Tests and CIs 11. Two-Sample Tests and CIs 12. Chi-Square Tests 13. Simple Regression 14. Regression Toward the Mean 15. Multiple Regression 16. Miscellaneous 17. Out-of-Class Projects
Welcome to the fascinating intersection of mathematics, biology, and ecology!
This book is intended primarily as a resource for teachers planning to teach their first introductory course on modeling in mathematical biology and/or ecology. This being said, it can also be used by students preparing to embark on an independent studies project in one of these fields; or, by researchers unfamiliar with the methods or software introduced who are seeking an accessible and quick introduction to one of the methods and/or software presented here; or, by curious biologists, ecologists, or mathematicians who may be unfamiliar with "the other side;" or, maybe, by the perpetual learner who is intrigued by the dynamics of living ecosystems. For each of the above, this book is designed to be an accessible introduction to the captivating landscape of biomathematics.
The approach used in this book takes advantage of technology in leading readers on a journey that bridges seemingly distinct fields through introductions to three methods and software platforms: Compartmental models with Berkeley Madonna; agent-based models with NetLogo; and cluster analysis through selforganizing maps using an R Shiny app.
This is not intended to be a textbook (though it may be used as one), nor is it a purely mathematics book or one purely about deeper aspects of biology or ecology. It focuses on three selected ways in which the intersection of mathematics and biology (and mathematics and ecology) can be explored with the help of software. Moreover, the manner in which the content is presented makes it possible to use this book to help prepare for an introductory course at a wide range of levels, depending on the discipline within which the course is taught and the mathematical prerequisites for the course.
There are four chapters, the first of which presents the reader with a bit of background information followed by suggestions on how to get the most out of this book. The three core chapters introduce the three previously mentioned methods and software in a manner envisioned to be accessible to most.
Introductory to intermediate level technology-based presentation.
Detailed examples of using technology followed by brief exploratory exercises.
Collections of applications along with resources that may be implemented using ideas presented.
This book is in the series
Volume 64 | Inverse and Ill-Posed Problems Series
This book provides a comprehensive exploration of Mean Field Games (MFG) theory, a mathematical framework for modeling the collective behavior of rational agents in complex systems. MFG theory can govern a range of societal phenomena, including finance, sociology, machine learning, and economics. The focus is on the system of two coupled nonlinear parabolic partial differential equations (PDEs) that define the Mean Field Games System. The book covers key theoretical topics such as solution stability and uniqueness, with a particular emphasis on Carleman estimates, which are used to estimate solution errors based on noise in the input data. It also introduces the theory of Ill-Posed and Inverse Problems within MFG theory. Both theoretical and numerical aspects of forward and inverse problems are explored through Carleman estimates, offering a rigorous foundation for researchers and practitioners in applied mathematics and related fields.
This book offers a rigorous approach to Carleman estimates, a key element of Mean Field Games theory, making it an essential resource for researchers, graduate students, and professionals looking to apply this powerful framework to complex, real-world systems.
Provides new results on stability and uniqueness for forward and inverse problems in the Mean Field Games System (MFGS).
Provides a novel tool for addressing challenges in the MFGS.
This book is in the series
Volume 102 | De Gruyter Studies in Mathematics
This volume applies theories of harmonic analysis to the study of nonlinear partial differential equations. It covers consolidation characterizations of differentiable function spaces, and the theory of three generations of C-Z singular integral operators, Fourier restriction estimation, Strichartz estimation, and Littlewood-Paley theory. It combines harmonic analysis methods with the study of partial differential equations.
Applies theories of harmonic analysis to the study of nonlinear PDEs using modern methods
Describes basic analytical tools in detail while systematically outlining frontier topics.
The self-contained format makes it ideal for self-study.