The stars of this book, vectors and tensors, are unlikely celebrities. If you ever took a physics course, the word vector might remind you of the mathematics needed to determine forces on an amusement park ride, a turbine, or a projectile. You might also remember that a vector is a quantity that has magnitude and (this is the key) direction. In fact, vectors are examples of tensors, which can represent even more data. It sounds simple enough--and yet, as award-winning science writer Robyn Arianrhod shows in this riveting story, the idea of a single symbol expressing more than one thing at once was millennia in the making. And without that idea, we wouldn't have such a deep understanding of our world.

Vector and tensor calculus offers an elegant language for expressing the way things behave in space and time, and Arianrhod shows how this enabled physicists and mathematicians to think in a brand-new way. These include James Clerk Maxwell when he ushered in the wireless electromagnetic age; Einstein when he predicted the curving of space-time and the existence of gravitational waves; Paul Dirac, when he created quantum field theory; and Emmy Noether, when she connected mathematical symmetry and the conservation of energy. For it turned out that it's not just physical quantities and dimensions that vectors and tensors can represent, but other dimensions and other kinds of information, too. This is why physicists and mathematicians can speak of four-dimensional space-time and other higher-dimensional spaces, and why you're likely relying on vectors or tensors whenever you use digital applications such as search engines, GPS, or your mobile phone.

In exploring the evolution of vectors and tensors--and introducing the fascinating people who gave them to us--Arianrhod takes readers on an extraordinary, five-thousand-year journey through the human imagination. She shows the genius required to reimagine the world--and how a clever mathematical construct can dramatically change discovery's direction.

The theoretical assumptions of the following mathematical topics are presented in this book:
vectors and vector calculus
matrices and matrix calculus
vector and matrix spaces
mathematics and tensor calculus

The aim of this book is to present a self-contained, reasonably modern account of tensor analysis and the calculus of exterior differential forms, adapted to the needs of physicists, engineers, and applied mathematicians. In the later, increasingly sophisticated chapters, the interaction between the concept of invariance and the calculus of variations is examined. This interaction is of profound importance to all physical field theories.
Beginning with simple physical examples, the theory of tensors and forms is developed by a process of successive abstractions. This enables the reader to infer generalized principles from concrete situations -- departing from the traditional approach to tensors and forms in terms of purely differential-geometric concepts.
The treatment of the calculus of variations of single and multiple integrals is based ab initio on Carath odory's method of equivalent integrals. Subsequent material explores the effects of invariance postulates on variational principles, focusing ultimately on relativistic field theories. Other discussions include:
- integral invariants
- simple and direct derivations of Noether's theorems
- Riemannian spaces with indefinite metrics
The emphasis in this book is on analytical techniques, with abundant problems, ranging from routine manipulative exercises to technically difficult problems encountered by those using tensor techniques in research activities. A special effort has been made to collect many useful results of a technical nature, not generally discussed in the standard literature. The Appendix, newly revised and enlarged for the Dover edition, presents a reformulation of the principal concepts of the main text within the terminology of current global differential geometry, thus bridging the gap between classical tensor analysis and the fundamentals of more recent global theories.

This is a first-rate book and deserves to be widely read. -- American Mathematical Monthly
Despite its success as a mathematical tool in the general theory of relativity and its adaptability to a wide range of mathematical and physical problems, tensor analysis has always had a rather restricted level of use, with an emphasis on notation and the manipulation of indices. This book is an attempt to broaden this point of view at the stage where the student first encounters the subject. The authors have treated tensor analysis as a continuation of advanced calculus, striking just the right balance between the formal and abstract approaches to the subject.
The material proceeds from the general to the special. An introductory chapter establishes notation and explains various topics in set theory and topology. Chapters 1 and 2 develop tensor analysis in its function-theoretical and algebraic aspects, respectively. The next two chapters take up vector analysis on manifolds and integration theory. In the last two chapters (5 and 6) several important special structures are studied, those in Chapter 6 illustrating how the previous material can be adapted to clarify the ideas of classical mechanics. The text as a whole offers numerous examples and problems.
A student with a background of advanced calculus and elementary differential equation could readily undertake the study of this book. The more mature the reader is in terms of other mathematical knowledge and experience, the more he will learn from this presentation.

This brilliant study by a famed mathematical scholar and former professor of mathematics at the University of Amsterdam integrates a concise exposition of the mathematical basis of tensor analysis with admirably chosen physical examples of the theory.
The first five chapters incisively set out the mathematical theory underlying the use of tensors. The tensor algebra in EN and RN is developed in Chapters I and II. Chapter II introduces a sub-group of the affine group, then deals with the identification of quantities in EN. The tensor analysis in XN is developed in Chapter IV. In chapters VI through IX, Professor Schouten presents applications of the theory that are both intrinsically interesting and good examples of the use and advantages of the calculus. Chapter VI, intimately connected with Chapter III, shows that the dimensions of physical quantities depend upon the choice of the underlying group, and that tensor calculus is the best instrument for dealing with the properties of anisotropic media. In Chapter VII, modern tensor calculus is applied to some old and some modern problems of elasticity and piezo-electricity. Chapter VIII presents examples concerning anholonomic systems and the homogeneous treatment of the equations of Lagrange and Hamilton. Chapter IX deals first with relativistic kinematics and dynamics, then offers an exposition of modern treatment of relativistic hydrodynamics. Chapter X introduces Dirac's matrix calculus. Two especially valuable features of the book are the exercises at the end of each chapter, and a summary of the mathematical theory contained in the first five chapters -- ideal for readers whose primary interest is in physics rather than mathematics.

This textbook for the undergraduate vector calculus course presents a unified treatment of vector and geometric calculus.

This is the printing of June 2024.

The book is a sequel to the text Linear and Geometric Algebra by the same author. That text is a prerequisite for this one. Its web page is at faculty.luther.edu/ macdonal/laga.

Linear algebra and vector calculus have provided the basic vocabulary of mathematics in dimensions greater than one for the past one hundred years. Just as geometric algebra generalizes linear algebra in powerful ways, geometric calculus generalizes vector calculus in powerful ways.

Traditional vector calculus topics are covered, as they must be, since readers will encounter them in other texts and out in the world.

Differential geometry is used today in many disciplines. A final chapter is devoted to it.

Download the book's table of contents, preface, and index at the book's web site: faculty.luther.edu/ macdonal/vagc.

From a review of Linear and Geometric Algebra:
Alan Macdonald's text is an excellent resource if you are just beginning the study of geometric algebra and would like to learn or review traditional linear algebra in the process. The clarity and evenness of the writing, as well as the originality of presentation that is evident throughout this text, suggest that the author has been successful as a mathematics teacher in the undergraduate classroom. This carefully crafted text is ideal for anyone learning geometric algebra in relative isolation, which I suspect will be the case for many readers.
-- Jeffrey Dunham, William R. Kenan Jr. Professor of Natural Sciences, Middlebury College

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De otro lado, la Matemática básica universitaria en especial la Introducción a la matemática para ingeniería, permite la formación del estudiante desde los primeros ciclos, en ese sentido, el Libro 2 - Parte III es el libro de Cálculo para ingenieros que permite continuar con ese proceso de formación. Contiene la siguiente sección:
Vectores en el espacio tridimensional. Recta y plano.

Cada sección con sus respectivos NOTEBOOK I y II. Los cuales permiten una autoevaluación y práctica constante, en todo momento dirigido por los autores, a través de las indicaciones y sugerencias que encontrará en los ejercicios. El objetivo es que usted compruebe, interiorice y confirme los conocimientos adquiridos. Disfrutará su aprendizaje como los primeros libros de Introducción a la matemática para ingeniería.

Comprende dos PARTES: Comunicación matemática y luego, el Modelamiento y resolución matemática.
El primero, contiene preguntas teóricas en diversas modalidades: para marcar verdadero o falso, con su respectiva justificación; preguntas para responder en forma concisa; preguntas abiertas; crear un ejercicio según indicaciones; demostración de teoremas; a partir de un gráfico o datos reconstruir el enunciado y por último, se presenta el ejercicio y se le pide responder en forma verbal (solo texto, tal como se describió en la Introducción al cálculo superior) sin escribir alguna fórmula y sin desarrollarlo, permitiéndole manejar un lenguaje matemático conocido como verbalización matemática (como aprender algún idioma extranjero). El segundo, contiene los ejercicios y problemas de aplicación propuestos, dirigidos en todo momento por los autores. Al final de cada NOTEBOOK tenemos una miscelánea, que son ejercicios que combinan diversos tópicos de la sección (requieren mayor dominio matemático).
VAMOS ES HORA DE DISFRUTAR LA TEMIDA MATEMÁTICA!
Este libro acaba de nacer, dale la oportunidad de vivir!

This contributed volume provides an extensive account of research and expository papers in a broad domain of mathematical analysis and its various applications to a multitude of fields. Presenting the state-of-the-art knowledge in a wide range of topics, the book will be useful to graduate students and researchers in theoretical and applicable interdisciplinary research. The focus is on several subjects including: optimal control problems, optimal maintenance of communication networks, optimal emergency evacuation with uncertainty, cooperative and noncooperative partial differential systems, variational inequalities and general equilibrium models, anisotropic elasticity and harmonic functions, nonlinear stochastic differential equations, operator equations, max-product operators of Kantorovich type, perturbations of operators, integral operators, dynamical systems involving maximal monotone operators, the three-body problem, deceptive systems, hyperbolic equations, strongly generalized preinvex functions, Dirichlet characters, probability distribution functions, applied statistics, integral inequalities, generalized convexity, global hyperbolicity of spacetimes, Douglas-Rachford methods, fixed point problems, the general Rodrigues problem, Banach algebras, affine group, Gibbs semigroup, relator spaces, sparse data representation, Meier-Keeler sequential contractions, hybrid contractions, and polynomial equations. Some of the works published within this volume provide as well guidelines for further research and proposals for new directions and open problems.