A Geometric Approach to Differential Forms

Author: David Bachman

Publisher: Springer Science & Business Media

ISBN: 0817683046

Category: Mathematics

Page: 156

View: 4855

This text presents differential forms from a geometric perspective accessible at the undergraduate level. It begins with basic concepts such as partial differentiation and multiple integration and gently develops the entire machinery of differential forms. The subject is approached with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually. Each new concept is presented with a natural picture that students can easily grasp. Algebraic properties then follow. The book contains excellent motivation, numerous illustrations and solutions to selected problems.
Posted in Mathematics

A Geometric Approach to Differential Forms

Author: David Bachman

Publisher: Springer Science & Business Media

ISBN: 0817645209

Category: Mathematics

Page: 133

View: 401

This text presents differential forms from a geometric perspective accessible at the undergraduate level. It begins with basic concepts such as partial differentiation and multiple integration and gently develops the entire machinery of differential forms. The subject is approached with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually. Each new concept is presented with a natural picture that students can easily grasp. Algebraic properties then follow. The book contains excellent motivation, numerous illustrations and solutions to selected problems.
Posted in Mathematics

A Geometric Approach to Differential Forms

Author: David Bachman

Publisher: Springer Science & Business Media

ISBN: 0817644997

Category: Mathematics

Page: 133

View: 8721

This text presents differential forms from a geometric perspective accessible at the undergraduate level. It begins with basic concepts such as partial differentiation and multiple integration and gently develops the entire machinery of differential forms. The subject is approached with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually. Each new concept is presented with a natural picture that students can easily grasp. Algebraic properties then follow. The book contains excellent motivation, numerous illustrations and solutions to selected problems.
Posted in Mathematics

Advanced Calculus

A Differential Forms Approach

Author: Harold M. Edwards

Publisher: Springer Science & Business Media

ISBN: 146120271X

Category: Mathematics

Page: 508

View: 2202

This book is a high-level introduction to vector calculus based solidly on differential forms. Informal but sophisticated, it is geometrically and physically intuitive yet mathematically rigorous. It offers remarkably diverse applications, physical and mathematical, and provides a firm foundation for further studies.
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A New Approach to Differential Geometry using Clifford's Geometric Algebra

Author: John Snygg

Publisher: Springer Science & Business Media

ISBN: 081768283X

Category: Mathematics

Page: 465

View: 6054

Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.
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Differential Forms

A Complement to Vector Calculus

Author: Steven H. Weintraub

Publisher: Academic Press

ISBN: 9780127425108

Category: Mathematics

Page: 256

View: 2158

This text is one of the first to treat vector calculus using differential forms in place of vector fields and other outdated techniques. Geared towards students taking courses in multivariable calculus, this innovative book aims to make the subject more readily understandable. Differential forms unify and simplify the subject of multivariable calculus, and students who learn the subject as it is presented in this book should come away with a better conceptual understanding of it than those who learn using conventional methods. * Treats vector calculus using differential forms * Presents a very concrete introduction to differential forms * Develops Stokess theorem in an easily understandable way * Gives well-supported, carefully stated, and thoroughly explained definitions and theorems. * Provides glimpses of further topics to entice the interested student
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Vector Calculus, Linear Algebra, and Differential Forms

A Unified Approach

Author: John H. Hubbard,Barbara Burke Hubbard

Publisher: N.A

ISBN: 9780136574460

Category: Mathematics

Page: 687

View: 9859

This text covers most of the standard topics in multivariate calculus and part of a standard first course in linear algebra. It focuses on underlying ideas, integrates theory and applications, offers a host of pedagogical aids, and features coverage of differential forms and an emphasis on numerical methods to prepare students for modern applications of mathematics. *Covers important material that is usually omitted. *Presents more difficult and longer proofs (e.g. Proofs of the Kantorovitch theorem, the implicit function theorem) in an appendix. *Makes a careful distinction between vectors and points. *Features an innovative approach to the implicit function theorem and inverse function theorem using Newton's method. *Always emphasizes the underlying meaning - what is really going on (generally, with a geometric interpretation) - eg. The chain rule is a composition of linear transformations; the point of the implicit function theorem is to guarantee that under certain circumstances, non-linear equations have solutions. *Integrates theory and applications. *Begins most chapters with a treatment of a linear problem and then shows how the 7 methods apply to corresponding non-linear p
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Applied Differential Geometry

Author: William L. Burke

Publisher: Cambridge University Press

ISBN: 9780521269292

Category: Mathematics

Page: 414

View: 4809

This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry. The intended audience is physicists, so the author emphasises applications and geometrical reasoning in order to give results and concepts a precise but intuitive meaning without getting bogged down in analysis. The large number of diagrams helps elucidate the fundamental ideas. Mathematical topics covered include differentiable manifolds, differential forms and twisted forms, the Hodge star operator, exterior differential systems and symplectic geometry. All of the mathematics is motivated and illustrated by useful physical examples.
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Differential Forms

Theory and Practice

Author: Steven H. Weintraub

Publisher: Elsevier

ISBN: 0123946174

Category: Mathematics

Page: 408

View: 7554

Differential forms are a powerful mathematical technique to help students, researchers, and engineers solve problems in geometry and analysis, and their applications. They both unify and simplify results in concrete settings, and allow them to be clearly and effectively generalized to more abstract settings. Differential Forms has gained high recognition in the mathematical and scientific community as a powerful computational tool in solving research problems and simplifying very abstract problems. Differential Forms, 2nd Edition, is a solid resource for students and professionals needing a general understanding of the mathematical theory and to be able to apply that theory into practice. Provides a solid theoretical basis of how to develop and apply differential forms to real research problems Includes computational methods to enable the reader to effectively use differential forms Introduces theoretical concepts in an accessible manner
Posted in Mathematics

Tensors, Differential Forms, and Variational Principles

Author: David Lovelock,Hanno Rund

Publisher: Courier Corporation

ISBN: 048613198X

Category: Mathematics

Page: 400

View: 9745

Incisive, self-contained account of tensor analysis and the calculus of exterior differential forms, interaction between the concept of invariance and the calculus of variations. Emphasis is on analytical techniques. Includes problems.
Posted in Mathematics

Exterior Analysis

Using Applications of Differential Forms

Author: Erdogan Suhubi

Publisher: Elsevier

ISBN: 0124159281

Category: Mathematics

Page: 779

View: 8653

Exterior analysis uses differential forms (a mathematical technique) to analyze curves, surfaces, and structures. Exterior Analysis is a first-of-its-kind resource that uses applications of differential forms, offering a mathematical approach to solve problems in defining a precise measurement to ensure structural integrity. The book provides methods to study different types of equations and offers detailed explanations of fundamental theories and techniques to obtain concrete solutions to determine symmetry. It is a useful tool for structural, mechanical and electrical engineers, as well as physicists and mathematicians. Provides a thorough explanation of how to apply differential equations to solve real-world engineering problems Helps researchers in mathematics, science, and engineering develop skills needed to implement mathematical techniques in their research Includes physical applications and methods used to solve practical problems to determine symmetry
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Geometry of Differential Forms

Author: Shigeyuki Morita

Publisher: American Mathematical Soc.

ISBN: 9780821810453

Category: Mathematics

Page: 321

View: 8940

Since the times of Gauss, Riemann, and Poincare, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms. This book is a comprehensive introduction to differential forms. It begins with a quick presentation of the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results about them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated in the book is a detailed description of the Chern-Weil theory. With minimal prerequisites, the book can serve as a textbook for an advanced undergraduate or a graduate course in differential geometry.
Posted in Mathematics

Clifford Algebra to Geometric Calculus

A Unified Language for Mathematics and Physics

Author: David Hestenes,Garret Sobczyk

Publisher: Springer Science & Business Media

ISBN: 9789027725615

Category: Mathematics

Page: 314

View: 7263

Geometric Calculus is a language for expressing and analyzing the full range of geometric concepts in mathematics. Clifford Algebra provides the grammar. Complex number, quaternions, matrix algebra, vector, tensor and spinor calculus and differential forms are integrated into a single comprehensive system. The geometric calculus developed in this book has the following features: a systematic development of definitions, concepts and theorems needed to apply the calculus easily and effectively to almost any branch of mathematics or physics; a formulation of linear algebra capable of details computations without matrices or coordinates; new proofs and treatments of canonical forms including an extensive discussion of spinor representations of rotations in Euclidean n-space; a new concept of differentiation which makes it possible to formulate calculus on manifolds and carry out complete calculations of such thinks as the Jacobian of a transformation without resorting to coordinates; a coordinate-free approach to differential geometry featuring a new quantity, the shape tensor, from which the curvature tensor can be computed without a connection; a formulation of integration theory based on a concept of directed measure, with new results including a generalization of Cauchy's integral formula to n-dimension spaces and explicit integral formula for the inverse of a transformation; a new approach to Lie groups and Lie algebras. --From cover.
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Advanced Calculus

A Geometric View

Author: James J. Callahan

Publisher: Springer Science & Business Media

ISBN: 9781441973320

Category: Mathematics

Page: 526

View: 1005

With a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus. Besides the classical capstones--the change of variables formula, implicit and inverse function theorems, the integral theorems of Gauss and Stokes--the text treats other important topics in differential analysis, such as Morse's lemma and the Poincaré lemma. The ideas behind most topics can be understood with just two or three variables. The book incorporates modern computational tools to give visualization real power. Using 2D and 3D graphics, the book offers new insights into fundamental elements of the calculus of differentiable maps. The geometric theme continues with an analysis of the physical meaning of the divergence and the curl at a level of detail not found in other advanced calculus books. This is a textbook for undergraduates and graduate students in mathematics, the physical sciences, and economics. Prerequisites are an introduction to linear algebra and multivariable calculus. There is enough material for a year-long course on advanced calculus and for a variety of semester courses--including topics in geometry. The measured pace of the book, with its extensive examples and illustrations, make it especially suitable for independent study.
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Differential Forms and the Geometry of General Relativity

Author: Tevian Dray

Publisher: CRC Press

ISBN: 1466510005

Category: Mathematics

Page: 321

View: 1486

Differential Forms and the Geometry of General Relativity provides readers with a coherent path to understanding relativity. Requiring little more than calculus and some linear algebra, it helps readers learn just enough differential geometry to grasp the basics of general relativity. The book contains two intertwined but distinct halves. Designed for advanced undergraduate or beginning graduate students in mathematics or physics, most of the text requires little more than familiarity with calculus and linear algebra. The first half presents an introduction to general relativity that describes some of the surprising implications of relativity without introducing more formalism than necessary. This nonstandard approach uses differential forms rather than tensor calculus and minimizes the use of "index gymnastics" as much as possible. The second half of the book takes a more detailed look at the mathematics of differential forms. It covers the theory behind the mathematics used in the first half by emphasizing a conceptual understanding instead of formal proofs. The book provides a language to describe curvature, the key geometric idea in general relativity.
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Differential Forms and Connections

Author: R. W. R. Darling

Publisher: Cambridge University Press

ISBN: 9780521468008

Category: Mathematics

Page: 256

View: 3944

This book introduces the tools of modern differential geometry--exterior calculus, manifolds, vector bundles, connections--and covers both classical surface theory, the modern theory of connections, and curvature. Also included is a chapter on applications to theoretical physics. The author uses the powerful and concise calculus of differential forms throughout. Through the use of numerous concrete examples, the author develops computational skills in the familiar Euclidean context before exposing the reader to the more abstract setting of manifolds. The only prerequisites are multivariate calculus and linear algebra; no knowledge of topology is assumed. Nearly 200 exercises make the book ideal for both classroom use and self-study for advanced undergraduate and beginning graduate students in mathematics, physics, and engineering.
Posted in Mathematics

Electrodynamics

A Modern Geometric Approach

Author: William Baylis

Publisher: Springer Science & Business Media

ISBN: 9780817640255

Category: Science

Page: 380

View: 6109

The emphasis in this text is on classical electromagnetic theory and electrodynamics, that is, dynamical solutions to the Lorentz-force and Maxwell's equations. The natural appearance of the Minkowski spacetime metric in the paravector space of Clifford's geometric algebra is used to formulate a covariant treatment in special relativity that seamlessly connects spacetime concepts to the spatial vector treatments common in undergraduate texts. Baylis' geometrical interpretation, using such powerful tools as spinors and projectors, essentially allows a component-free notation and avoids the clutter of indices required in tensorial treatments. The exposition is clear and progresses systematically - from a discussion of electromagnetic units and an explanation of how the SI system can be readily converted to the Gaussian or natural Heaviside-Lorentz systems, to an introduction of geometric algebra and the paravector model of spacetime, and finally, special relativity. Other topics include Maxwell's equation(s), the Lorentz-force law, the Fresnel equations, electromagnetic waves and polarization, wave guides, radiation from accelerating charges and time-dependent currents, the Liénard-Wiechert potentials, and radiation reaction, all of which benefit from the modern relativistic approach. Numerous worked examples and exercises dispersed throughout the text help the reader understand new concepts and facilitate self-study of the material. Each chapter concludes with a set of problems, many with answers. Complete solutions are also available. An excellent feature is the integration of Maple into the text, thereby facilitating difficult calculations. To download accompanying Maple worksheets, please visit http://www.cs.uwindsor.ca/users/b/baylis
Posted in Science

Elementary Differential Geometry

Author: Barrett O'Neill

Publisher: Academic Press

ISBN: 148326811X

Category: Mathematics

Page: 422

View: 1688

Elementary Differential Geometry focuses on the elementary account of the geometry of curves and surfaces. The book first offers information on calculus on Euclidean space and frame fields. Topics include structural equations, connection forms, frame fields, covariant derivatives, Frenet formulas, curves, mappings, tangent vectors, and differential forms. The publication then examines Euclidean geometry and calculus on a surface. Discussions focus on topological properties of surfaces, differential forms on a surface, integration of forms, differentiable functions and tangent vectors, congruence of curves, derivative map of an isometry, and Euclidean geometry. The manuscript takes a look at shape operators, geometry of surfaces in E, and Riemannian geometry. Concerns include geometric surfaces, covariant derivative, curvature and conjugate points, Gauss-Bonnet theorem, fundamental equations, global theorems, isometries and local isometries, orthogonal coordinates, and integration and orientation. The text is a valuable reference for students interested in elementary differential geometry.
Posted in Mathematics

Differential Forms for Cartan-Klein Geometry

Author: Jose G. Vargas

Publisher: Abramis

ISBN: 9781845495299

Category: Science

Page: 270

View: 7792

This book lets readers understand differential geometry with differential forms. It is unique in providing detailed treatments of topics not normally found elsewhere, like the programs of B. Riemann and F. Klein in the second half of the 19th century, and their being superseded by E. Cartan in the twentieth. Several conservation laws are presented in a unified way. The Einstein 3-form rather than the Einstein tensor is emphasized; their relationship is shown. Examples are chosen for their pedagogic value. Numerous advanced comments are sprinkled throughout the text. The equations of structure are addressed in different ways. First, in affine and Euclidean spaces, where torsion and curvature simply happen to be zero. In a second approach, the 2-torus and the punctured plane and 2-sphere are endowed with the "Columbus connection," torsion becoming a concept which could have been understood even by sailors of the 15th century. Those equations are then presented as the breaking of integrability conditions for connection equations. Finally, a topological definition brings together the concepts of connection and equations of structure. These options should meet the needs and learning objectives of readers with very different backgrounds. Dr Howard E Brandt
Posted in Science

Linear Algebra

A Geometric Approach

Author: Ted Shifrin,Malcolm Adams

Publisher: Macmillan

ISBN: 1429215216

Category: Mathematics

Page: 464

View: 7077

Linear Algebra: A Geometric Approach, Second Edition, is a text that not only presents the standard computational aspects of linear algebra and interesting applications, it guides students to think about mathematical concepts and write rigorous mathematical arguments. This thought-provoking introduction to the subject and its myriad applications is interesting to the science or engineering student but will also help the mathematics student make the transition to more abstract advanced courses. The second edition has been updated with additional examples and exercises and has been streamlined for easier teaching and studying.
Posted in Mathematics