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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Integral equations

Probability Theory - A Comprehensive Course (Paperback, 3rd ed. 2020): Achim Klenke Probability Theory - A Comprehensive Course (Paperback, 3rd ed. 2020)
Achim Klenke
R1,600 R1,464 Discovery Miles 14 640 Save R136 (8%) Ships in 9 - 15 working days

This popular textbook, now in a revised and expanded third edition, presents a comprehensive course in modern probability theory.Probability plays an increasingly important role not only in mathematics, but also in physics, biology, finance and computer science, helping to understand phenomena such as magnetism, genetic diversity and market volatility, and also to construct efficient algorithms. Starting with the very basics, this textbook covers a wide variety of topics in probability, including many not usually found in introductory books, such as: limit theorems for sums of random variables martingales percolation Markov chains and electrical networks construction of stochastic processes Poisson point process and infinite divisibility large deviation principles and statistical physics Brownian motion stochastic integrals and stochastic differential equations. The presentation is self-contained and mathematically rigorous, with the material on probability theory interspersed with chapters on measure theory to better illustrate the power of abstract concepts. This third edition has been carefully extended and includes new features, such as concise summaries at the end of each section and additional questions to encourage self-reflection, as well as updates to the figures and computer simulations. With a wealth of examples and more than 290 exercises, as well as biographical details of key mathematicians, it will be of use to students and researchers in mathematics, statistics, physics, computer science, economics and biology.

Lectures on Convex Geometry (Paperback, 1st ed. 2020): Daniel Hug, Wolfgang Weil Lectures on Convex Geometry (Paperback, 1st ed. 2020)
Daniel Hug, Wolfgang Weil
R1,387 Discovery Miles 13 870 Ships in 9 - 15 working days

This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn-Minkowski theory, with an exposition of mixed volumes, the Brunn-Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.

Lebesgue's Theory of Integration - Its Origins and Development (Hardcover, New edition): Lebesgue's Theory of Integration - Its Origins and Development (Hardcover, New edition)
R1,831 Discovery Miles 18 310 Ships in 12 - 17 working days

In this book, Hawkins elegantly places Lebesgue's early work on integration theory within in proper historical context by relating it to the developments during the nineteenth century that motivated it and gave it significance and also to the contributions made in this field by Lebesgue's contemporaries. Hawkins was awarded the 1997 MAA Chauvenet Prize and the 2001 AMS Albert Leon Whiteman Memorial Prize for notable exposition and exceptional scholarship in the history of mathematics.

Lectures on Ergodic Theory (Paperback): Paul R. Halmos Lectures on Ergodic Theory (Paperback)
Paul R. Halmos
R356 R321 Discovery Miles 3 210 Save R35 (10%) Ships in 9 - 15 working days
Line Integral Methods for Conservative Problems (Hardcover): Luigi Brugnano, Felice  Iavernaro Line Integral Methods for Conservative Problems (Hardcover)
Luigi Brugnano, Felice Iavernaro
R5,313 Discovery Miles 53 130 Ships in 12 - 17 working days

Line Integral Methods for Conservative Problems explains the numerical solution of differential equations within the framework of geometric integration, a branch of numerical analysis that devises numerical methods able to reproduce (in the discrete solution) relevant geometric properties of the continuous vector field. The book focuses on a large set of differential systems named conservative problems, particularly Hamiltonian systems. Assuming only basic knowledge of numerical quadrature and Runge-Kutta methods, this self-contained book begins with an introduction to the line integral methods. It describes numerous Hamiltonian problems encountered in a variety of applications and presents theoretical results concerning the main instance of line integral methods: the energy-conserving Runge-Kutta methods, also known as Hamiltonian boundary value methods (HBVMs). The authors go on to address the implementation of HBVMs in order to recover in the numerical solution what was expected from the theory. The book also covers the application of HBVMs to handle the numerical solution of Hamiltonian partial differential equations (PDEs) and explores extensions of the energy-conserving methods. With many examples of applications, this book provides an accessible guide to the subject yet gives you enough details to allow concrete use of the methods. MATLAB codes for implementing the methods are available online.

Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions (Paperback): Thomas... Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions (Paperback)
Thomas Trogdon, Sheehan Olver
R2,791 Discovery Miles 27 910 Ships in 12 - 17 working days

Riemann-Hilbert problems are fundamental objects of study within complex analysis. Many problems in differential equations and integrable systems, probability and random matrix theory, and asymptotic analysis can be solved by reformulation as a Riemann-Hilbert problem. This book, the most comprehensive one to date on the applied and computational theory of Riemann-Hilbert problems, includes:* An introduction to computational complex analysis.* An introduction to the applied theory of Riemann-Hilbert problems from an analytical and numerical perspective.* A discussion of applications to integrable systems, differential equations, and special function theory.* Six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann-Hilbert method, each of mathematical or physical significance or both.

Meromorphic Dynamics: Volume 1 - Abstract Ergodic Theory, Geometry, Graph Directed Markov Systems, and Conformal Measures... Meromorphic Dynamics: Volume 1 - Abstract Ergodic Theory, Geometry, Graph Directed Markov Systems, and Conformal Measures (Hardcover)
Janina Kotus, Mariusz Urbanski
R3,435 Discovery Miles 34 350 Ships in 12 - 17 working days

This text, the first of two volumes, provides a comprehensive and self-contained introduction to a wide range of fundamental results from ergodic theory and geometric measure theory. Topics covered include: finite and infinite abstract ergodic theory, Young's towers, measure-theoretic Kolmogorov-Sinai entropy, thermodynamics formalism, geometric function theory, various kinds of conformal measures, conformal graph directed Markov systems and iterated functions systems, semi-local dynamics of analytic functions, and nice sets. Many examples are included, along with detailed explanations of essential concepts and full proofs, in what is sure to be an indispensable reference for both researchers and graduate students.

Differential and Integral Calculus V 2 (Paperback, Wiley Classics): R. Courant Differential and Integral Calculus V 2 (Paperback, Wiley Classics)
R. Courant
R5,399 Discovery Miles 53 990 Ships in 10 - 15 working days

Partial table of contents:

Preliminary Remarks on Analytical Geometry and Vector Analysis: Rectangular Coordinates and Vectors, Affine Transformations and the Multiplication of Determinants.

Functions of Several Variables and Their Derivatives: Continuity, The Total Differential of a Function and Its Geometrical Meaning.

Developments and Applications of the Differential Calculus: Implicit Functions, Maxima and Minima.

Multiple Integrals: Transformation of Multiple Integrals, Improper Integrals.

Integration over Regions in Several Dimensions: Surface Integrals, Stokes's Theorem in Space.

Differential Equations: Examples on the Mechanics of a Particle, Linear Differential Equations.

Calculus of Variations: Euler's Differential Equation in the Simplest Case, Generalizations.

Functions of a Complex Variable: The Integration of Analytic Functions, Cauchy's Formula and Its Applications.

Appendixes.

Index.

Topics in Integration Research (Hardcover): Mark Burgin Topics in Integration Research (Hardcover)
Mark Burgin
R5,267 Discovery Miles 52 670 Ships in 12 - 17 working days

In calculus, we integrate functions using two types of integration -- definite integration and indefinite integration. In functional analysis, we integrate operators. To find a solution of a differential equation, we integrate this equation. Going beyond mathematics, we see that in databases, we integrate data, as well as database schemas. In electronics, integrated circuits have become central components of computers, calculators, cellular phones, and other digital appliances, which are now inextricable parts of the structure of modern societies. In economics, we have integration of the economy of one country into the economy of a union of other countries, eg: integration of economy of Hungary into the European Union economy. There is political integration and there is social integration. Thus, we can see many types and kinds of integration. Design of complex database schemas is based on a gradual integration of external schemas. Research presented in this book studies integration in mathematics and its applications. However, it is not only classical integration of functions but also fuzzy integration, integration of structures, probability as integration of random characteristics and integral operators in bundles with a hyperspace base.

Complex Integration - A Compendium of Smart and Little-Known Techniques for Evaluating Integrals and Sums (Paperback, 1st ed.... Complex Integration - A Compendium of Smart and Little-Known Techniques for Evaluating Integrals and Sums (Paperback, 1st ed. 2023)
Ron Gordon
R2,119 Discovery Miles 21 190 Ships in 10 - 15 working days

Integrals and sums are not generally considered for evaluation using complex integration. This book proposes techniques that mainly use complex integration and are quite different from those in the existing texts. Such techniques, ostensibly taught in Complex Analysis courses to undergraduate students who have had two semesters of calculus, are usually limited to a very small set of problems. Few practitioners consider complex integration as a tool for computing difficult integrals. While there are a number of books on the market that provide tutorials on this subject, the existing texts in this field focus on real methods. Accordingly, this book offers an eye-opening experience for computation enthusiasts used to relying on clever substitutions and transformations to evaluate integrals and sums. The book is the result of nine years of providing solutions to difficult calculus problems on forums such as Math Stack Exchange or the author's website, residuetheorem.com. It serves to detail to the enthusiastic mathematics undergraduate, or the physics or engineering graduate student, the art and science of evaluating difficult integrals, sums, and products.

Meromorphic Dynamics: Volume 2 - Elliptic Functions with an Introduction to the Dynamics of Meromorphic Functions (Hardcover):... Meromorphic Dynamics: Volume 2 - Elliptic Functions with an Introduction to the Dynamics of Meromorphic Functions (Hardcover)
Janina Kotus, Mariusz Urbanski
R3,716 Discovery Miles 37 160 Ships in 12 - 17 working days

This text, the second of two volumes, builds on the foundational material on ergodic theory and geometric measure theory provided in Volume I, and applies all the techniques discussed to describe the beautiful and rich dynamics of elliptic functions. The text begins with an introduction to topological dynamics of transcendental meromorphic functions, before progressing to elliptic functions, discussing at length their classical properties, measurable dynamics and fractal geometry. The authors then look in depth at compactly non-recurrent elliptic functions. Much of this material is appearing for the first time in book or paper form. Both senior and junior researchers working in ergodic theory and dynamical systems will appreciate what is sure to be an indispensable reference.

Perspectives in Dynamical Systems I: Mechatronics and Life Sciences - DSTA, Lodz, Poland December 2-5, 2019 (Paperback, 1st ed.... Perspectives in Dynamical Systems I: Mechatronics and Life Sciences - DSTA, Lodz, Poland December 2-5, 2019 (Paperback, 1st ed. 2022)
Jan Awrejcewicz
R5,087 Discovery Miles 50 870 Ships in 10 - 15 working days

This volume is part of collection of contributions devoted to analytical and experimental techniques of dynamical systems, presented at the 15th International Conference "Dynamical Systems: Theory and Applications", held in Lodz, Poland on December 2-5, 2019. The wide selection of material has been divided into three volumes, each focusing on a different field of applications of dynamical systems. The broadly outlined focus of both the conference and these books includes bifurcations and chaos in dynamical systems, asymptotic methods in nonlinear dynamics, dynamics in life sciences and bioengineering, original numerical methods of vibration analysis, control in dynamical systems, optimization problems in applied sciences, stability of dynamical systems, experimental and industrial studies, vibrations of lumped and continuous systems, non-smooth systems, engineering systems and differential equations, mathematical approaches to dynamical systems, and mechatronics.

Perspectives in Dynamical Systems III: Control and Stability - DSTA, Lodz, Poland December 2-5, 2019 (Paperback, 1st ed. 2021):... Perspectives in Dynamical Systems III: Control and Stability - DSTA, Lodz, Poland December 2-5, 2019 (Paperback, 1st ed. 2021)
Jan Awrejcewicz
R5,109 Discovery Miles 51 090 Ships in 10 - 15 working days

This volume is part of collection of contributions devoted to analytical and experimental techniques of dynamical systems, presented at the 15th International Conference "Dynamical Systems: Theory and Applications", held in Lodz, Poland on December 2-5, 2019. The wide selection of material has been divided into three volumes, each focusing on a different field of applications of dynamical systems. The broadly outlined focus of both the conference and these books includes bifurcations and chaos in dynamical systems, asymptotic methods in nonlinear dynamics, dynamics in life sciences and bioengineering, original numerical methods of vibration analysis, control in dynamical systems, optimization problems in applied sciences, stability of dynamical systems, experimental and industrial studies, vibrations of lumped and continuous systems, non-smooth systems, engineering systems and differential equations, mathematical approaches to dynamical systems, and mechatronics.

Modern Methods in Mathematical Physics - Integral Equations in Wolfram Mathematica (Paperback, 1st ed. 2022): Vladimir Ryzhov,... Modern Methods in Mathematical Physics - Integral Equations in Wolfram Mathematica (Paperback, 1st ed. 2022)
Vladimir Ryzhov, Tatiana Fedorova, Kirill Safronov, Shaharin Anwar Sulaiman, Samsul Ariffin Abdul Karim
R1,495 Discovery Miles 14 950 Ships in 10 - 15 working days

This book provides ideas for implementing Wolfram Mathematica to solve linear integral equations. The book introduces necessary theoretical information about exact and numerical methods of solving integral equations. Every method is supplied with a large number of detailed solutions in Wolfram Mathematica. In addition, the book includes tasks for individual study. This book is a supplement for students studying "Integral Equations". In addition, the structure of the book with individual assignments allows to use it as a base for various courses.

A View from the Top - Analysis, Combinatorics and Number Theory (Paperback, UK ed.): A View from the Top - Analysis, Combinatorics and Number Theory (Paperback, UK ed.)
R1,567 Discovery Miles 15 670 Ships in 12 - 17 working days

This book is based on a capstone course that the author taught to upper division undergraduate students with the goal to explain and visualize the connections between different areas of mathematics and the way different subject matters flow from one another. In teaching his readers a variety of problem solving techniques as well, the author succeeds in enhancing the readers' hands on knowledge of mathematics and provides glimpses into the world of research and discovery. The connections between different techniques and areas of mathematics are emphasized throughout and constitute one of the most important lessons this book attempts to impart. This book is interesting and accessible to anyone with a basic knowledge of high school mathematics and a curiosity about research mathematics. The author is a professor at the University of Missouri and has maintained a keen interest in teaching at different levels since his undergraduate days at the University of Chicago. He has run numerous summer programs in mathematics for local high school students and undergraduate students at his university.The author gets much of his research inspiration from his teaching activities and looks forward to exploring this wonderful and rewarding symbiosis for years to come.

Operator Theory and Harmonic Analysis - OTHA 2020, Part I - New General Trends and Advances of the Theory (Paperback, 1st ed.... Operator Theory and Harmonic Analysis - OTHA 2020, Part I - New General Trends and Advances of the Theory (Paperback, 1st ed. 2021)
Alexey N. Karapetyants, Vladislav V. Kravchenko, Elijah Liflyand, Helmuth R. Malonek
R6,404 Discovery Miles 64 040 Ships in 10 - 15 working days

This volume is part of the collaboration agreement between Springer and the ISAAC society. This is the first in the two-volume series originating from the 2020 activities within the international scientific conference "Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis" (OTHA), Southern Federal University in Rostov-on-Don, Russia. This volume is focused on general harmonic analysis and its numerous applications. The two volumes cover new trends and advances in several very important fields of mathematics, developed intensively over the last decade. The relevance of this topic is related to the study of complex multiparameter objects required when considering operators and objects with variable parameters.

Applications of Wavelet Multiresolution Analysis (Paperback, 1st ed. 2021): Juan Pablo Muszkats, Silvia Alejandra Seminara,... Applications of Wavelet Multiresolution Analysis (Paperback, 1st ed. 2021)
Juan Pablo Muszkats, Silvia Alejandra Seminara, Maria Ines Troparevsky
R2,817 Discovery Miles 28 170 Ships in 10 - 15 working days

This work results from a selection of the contributions presented in the mini symposium "Applications of Multiresolution Analysis with "Wavelets", presented at the ICIAM 19, the International Congress on Industrial and Applied Mathematics held at Valencia, Spain, in July 2019. The presented developments and applications cover different areas, including filtering, signal analysis for damage detection, time series analysis, solutions to boundary value problems and fractional calculus. This bunch of examples highlights the importance of multiresolution analysis to face problems in several and varied disciplines. The book is addressed to researchers in the field.

Inverse Acoustic and Electromagnetic Scattering Theory (Paperback, 4th ed. 2019): David Colton, Rainer Kress Inverse Acoustic and Electromagnetic Scattering Theory (Paperback, 4th ed. 2019)
David Colton, Rainer Kress
R3,373 R3,193 Discovery Miles 31 930 Save R180 (5%) Ships in 9 - 15 working days

The inverse scattering problem is central to many areas of science and technology such as radar, sonar, medical imaging, geophysical exploration and nondestructive testing. This book is devoted to the mathematical and numerical analysis of the inverse scattering problem for acoustic and electromagnetic waves. In this fourth edition, a number of significant additions have been made including a new chapter on transmission eigenvalues and a new section on the impedance boundary condition where particular attention has been made to the generalized impedance boundary condition and to nonlocal impedance boundary conditions. Brief discussions on the generalized linear sampling method, the method of recursive linearization, anisotropic media and the use of target signatures in inverse scattering theory have also been added.

Operator Theory and Harmonic Analysis - OTHA 2020, Part II - Probability-Analytical Models, Methods and Applications... Operator Theory and Harmonic Analysis - OTHA 2020, Part II - Probability-Analytical Models, Methods and Applications (Paperback, 1st ed. 2021)
Alexey N. Karapetyants, Igor V. Pavlov, Albert N. Shiryaev
R5,616 Discovery Miles 56 160 Ships in 10 - 15 working days

This volume is part of the collaboration agreement between Springer and the ISAAC society. This is the second in the two-volume series originating from the 2020 activities within the international scientific conference "Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis" (OTHA), Southern Federal University, Rostov-on-Don, Russia. This volume focuses on mathematical methods and applications of probability and statistics in the context of general harmonic analysis and its numerous applications. The two volumes cover new trends and advances in several very important fields of mathematics, developed intensively over the last decade. The relevance of this topic is related to the study of complex multi-parameter objects required when considering operators and objects with variable parameters.

Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity (Paperback, 1st ed. 2020): Alexander A. Lokshin Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity (Paperback, 1st ed. 2020)
Alexander A. Lokshin
R1,479 Discovery Miles 14 790 Ships in 10 - 15 working days

The objective of this book is to construct a rigorous mathematical approach to linear hereditary problems of wave propagation theory and demonstrate the efficiency of mathematical theorems in hereditary mechanics. By using both real end complex Tauberian techniques for the Laplace transform, a classification of near-front asymptotics of solutions to considered equations is given-depending on the singularity character of the memory function. The book goes on to derive the description of the behavior of these solutions and demonstrates the importance of nonlinear Laplace transform in linear hereditary elasticity. This book is of undeniable value to researchers working in areas of mathematical physics and related fields.

Stochastic Analysis (Paperback, 1st ed. 2020): Shigeo Kusuoka Stochastic Analysis (Paperback, 1st ed. 2020)
Shigeo Kusuoka
R3,347 Discovery Miles 33 470 Ships in 10 - 15 working days

This book is intended for university seniors and graduate students majoring in probability theory or mathematical finance. In the first chapter, results in probability theory are reviewed. Then, it follows a discussion of discrete-time martingales, continuous time square integrable martingales (particularly, continuous martingales of continuous paths), stochastic integrations with respect to continuous local martingales, and stochastic differential equations driven by Brownian motions. In the final chapter, applications to mathematical finance are given. The preliminary knowledge needed by the reader is linear algebra and measure theory. Rigorous proofs are provided for theorems, propositions, and lemmas. In this book, the definition of conditional expectations is slightly different than what is usually found in other textbooks. For the Doob-Meyer decomposition theorem, only square integrable submartingales are considered, and only elementary facts of the square integrable functions are used in the proof. In stochastic differential equations, the Euler-Maruyama approximation is used mainly to prove the uniqueness of martingale problems and the smoothness of solutions of stochastic differential equations.

New Trends on Analysis and Geometry in Metric Spaces - Levico Terme, Italy 2017 (Paperback, 1st ed. 2022): Luigi Ambrosio,... New Trends on Analysis and Geometry in Metric Spaces - Levico Terme, Italy 2017 (Paperback, 1st ed. 2022)
Luigi Ambrosio, Bruno Franchi, Irina Markina, Francesco Serra Cassano; Fabrice Baudoin, …
R1,651 Discovery Miles 16 510 Ships in 10 - 15 working days

This book includes four courses on geometric measure theory, the calculus of variations, partial differential equations, and differential geometry. Authored by leading experts in their fields, the lectures present different approaches to research topics with the common background of a relevant underlying, usually non-Riemannian, geometric structure. In particular, the topics covered concern differentiation and functions of bounded variation in metric spaces, Sobolev spaces, and differential geometry in the so-called Carnot-Caratheodory spaces. The text is based on lectures presented at the 10th School on "Analysis and Geometry in Metric Spaces" held in Levico Terme (TN), Italy, in collaboration with the University of Trento, Fondazione Bruno Kessler and CIME, Italy. The book is addressed to both graduate students and researchers.

Mittag-Leffler Functions, Related Topics and Applications (Paperback, 2nd ed. 2020): Rudolf Gorenflo, Anatoly A. Kilbas,... Mittag-Leffler Functions, Related Topics and Applications (Paperback, 2nd ed. 2020)
Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei Rogosin
R4,179 Discovery Miles 41 790 Ships in 10 - 15 working days

The 2nd edition of this book is essentially an extended version of the 1st and provides a very sound overview of the most important special functions of Fractional Calculus. It has been updated with material from many recent papers and includes several surveys of important results known before the publication of the 1st edition, but not covered there. As a result of researchers' and scientists' increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have caught the interest of the scientific community. Focusing on the theory of Mittag-Leffler functions, this volume offers a self-contained, comprehensive treatment, ranging from rather elementary matters to the latest research results. In addition to the theory the authors devote some sections of the work to applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular, the Mittag-Leffler functions make it possible to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and related special functions. The book is intended for a broad audience, comprising graduate students, university instructors and scientists in the field of pure and applied mathematics, as well as researchers in applied sciences like mathematical physics, theoretical chemistry, bio-mathematics, control theory and several other related areas.

Set-Valued Stochastic Integrals and Applications (Paperback, 1st ed. 2020): Michal Kisielewicz Set-Valued Stochastic Integrals and Applications (Paperback, 1st ed. 2020)
Michal Kisielewicz
R3,120 Discovery Miles 31 200 Ships in 10 - 15 working days

This book is among the first concise presentations of the set-valued stochastic integration theory as well as its natural applications, as well as the first to contain complex approach theory of set-valued stochastic integrals. Taking particular consideration of set-valued Ito , set-valued stochastic Lebesgue, and stochastic Aumann integrals, the volume is divided into nine parts. It begins with preliminaries of mathematical methods that are then applied in later chapters containing the main results and some of their applications, and contains many new problems. Methods applied in the book are mainly based on functional analysis, theory of probability processes, and theory of set-valued mappings. The volume will appeal to students of mathematics, economics, and engineering, as well as to mathematics professionals interested in applications of the theory of set-valued stochastic integrals.

Geometric Multivector Analysis - From Grassmann to Dirac (Hardcover, 1st ed. 2019): Andreas Rosen Geometric Multivector Analysis - From Grassmann to Dirac (Hardcover, 1st ed. 2019)
Andreas Rosen
R2,054 Discovery Miles 20 540 Ships in 9 - 15 working days

This book presents a step-by-step guide to the basic theory of multivectors and spinors, with a focus on conveying to the reader the geometric understanding of these abstract objects. Following in the footsteps of M. Riesz and L. Ahlfors, the book also explains how Clifford algebra offers the ideal tool for studying spacetime isometries and Moebius maps in arbitrary dimensions. The book carefully develops the basic calculus of multivector fields and differential forms, and highlights novelties in the treatment of, e.g., pullbacks and Stokes's theorem as compared to standard literature. It touches on recent research areas in analysis and explains how the function spaces of multivector fields are split into complementary subspaces by the natural first-order differential operators, e.g., Hodge splittings and Hardy splittings. Much of the analysis is done on bounded domains in Euclidean space, with a focus on analysis at the boundary. The book also includes a derivation of new Dirac integral equations for solving Maxwell scattering problems, which hold promise for future numerical applications. The last section presents down-to-earth proofs of index theorems for Dirac operators on compact manifolds, one of the most celebrated achievements of 20th-century mathematics. The book is primarily intended for graduate and PhD students of mathematics. It is also recommended for more advanced undergraduate students, as well as researchers in mathematics interested in an introduction to geometric analysis.

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