Vector calculus, linear algebra, and differential forms : a unified approach / John Hamal Hubbard, Barbara Burke Hubbard.

Format:

Book

Author/Creator:

Hubbard, John H. (John Hamal), 1945 or 1946-

Contributor:

Hubbard, Barbara Burke, 1948-

Classes of 1883 and 1884 Fund.

Language:

English

Subjects (All):

Calculus.

Algebras, Linear.

Physical Description:

xvi, 800 pages : illustrations ; 25 cm

Edition:

Second edition.

Place of Publication:

Upper Saddle River, NJ : Prentice Hall, [2002]

Summary:

Using a dual presentation that is rigorous and comprehensive--yet "exceptionaly reader-friendly" in approach--this book covers most of the standard topics in multivariate calculus and an introduction to linear algebra. It focuses in underlying ideas, integrates theory and applications, offers a host of learning aids, features coverage of differential forms, and emphasizes numerical methods that highlight modern applications of mathematics. The revised and expanded content of this edition includes new discussions of functions; complex numbers; closure, interior, and boundary; orientation; forms restricted to vector spaces; expanded discussions of subsets and subspaces of R DEGREESn"; probability, change of basis matrix; and more. For individuals interested in the fields of mathematics, engineering, and science--and looking for a unified approach and better understanding of vector calculus, linear algebra, and differential forms.

Contents:

0.1 Reading Mathematics 1

0.2 Quantifiers and Negation 4

0.3 Set Theory 6

0.4 Functions 9

0.5 Real Numbers 17

0.6 Infinite Sets 22

0.7 Complex Numbers 26

Chapter 1 Vectors, Matrices, and Derivatives

1.1 Introducing the Actors: Points and Vectors 34

1.2 Introducing the Actors: Matrices 43

1.3 A Matrix as a Transformation 59

1.4 The Geometry of R[superscript n] 71

1.5 Limits and Continuity 89

1.6 Four Big Theorems 110

1.7 Differential Calculus 125

1.8 Rules for Computing Derivatives 146

1.9 Mean Value Theorem and Criteria for Differentiability 154

Chapter 2 Solving Equations

2.1 The Main Algorithm: Row Reduction 170

2.2 Solving Equations Using Row Reduction 178

2.3 Matrix Inverses and Elementary Matrices 186

2.4 Linear Combinations, Span, and Linear Independence 192

2.5 Kernels, Images, and the Dimension Formula 206

2.6 An Introduction to Abstract Vector Spaces 224

2.7 Newton's Method 237

2.8 Superconvergence 257

2.9 The Inverse and Implicit Function Theorems 264

Chapter 3 Higher Partial Derivatives, Quadratic Forms, and Manifolds

3.1 Manifolds 292

3.2 Tangent Spaces 316

3.3 Taylor Polynomials in Several Variables 323

3.4 Rules for Computing Taylor Polynomials 335

3.5 Quadratic Forms 343

3.6 Classifying Critical Points of Functions 353

3.7 Constrained Critical Points and Lagrange Multipliers 360

3.8 Geometry of Curves and Surfaces 377

Chapter 4 Integration

4.1 Defining the Integral 400

4.2 Probability and Centers of Gravity 415

4.3 What Functions Can Be Integrated? 428

4.4 Integration and Measure Zero (Optional) 435

4.5 Fubini's Theorem and Iterated Integrals 443

4.6 Numerical Methods of Integration 455

4.7 Other Pavings 467

4.8 Determinants 469

4.9 Volumes and Determinants 485

4.10 The Change of Variables Formula 492

4.11 Lebesgue Integrals 505

Chapter 5 Volumes of Manifolds

5.1 Parallelograms and their Volumes 528

5.2 Parametrizations 532

5.3 Computing Volumes of Manifolds 540

5.4 Fractals and Fractional Dimension 553

Chapter 6 Forms and Vector Calculus

6.1 Forms on R[superscript n] 558

6.2 Integrating Form Fields over Parametrized Domains 574

6.3 Orientation of Manifolds 579

6.4 Integrating Forms over Oriented Manifolds 590

6.5 Forms and Vector Calculus 602

6.6 Boundary Orientation 614

6.7 The Exterior Derivative 627

6.8 The Exterior Derivative in the Language of Vector Calculus 635

6.9 The Generalized Stokes's Theorem 642

6.10 The Integral Theorems of Vector Calculus 651

6.11 Potentials 658

Appendix A Some Harder Proofs

A.1 Arithmetic of Real Numbers 669

A.2 Cubic and Quartic Equations 673

A.3 Two Extra Results in Topology 679

A.4 Proof of the Chain Rule 680

A.5 Proof of Kantorovich's theorem 682

A.6 Proof of Lemma 2.8.5 (Superconvergence) 688

A.7 Proof of Differentiability of the Inverse Function 690

A.8 Proof of the Implicit Function Theorem 693

A.9 Proof of Theorem 3.3.9: Equality of Crossed Partials 696

A.10 Proof of Proposition 3.3.19 698

A.11 Proof of Rules for Taylor Polynomials 701

A.12 Taylor's Theorem with Remainder 706

A.13 Proof of Theorem 3.5.3 (Completing Squares) 711

A.14 Geometry of Curves and Surfaces: Proofs 712

A.15 Proof of the Central Limit Theorem 718

A.16 Proof of Fubini's Theorem 722

A.17 Justifying the Use of Other Pavings 726

A.18 Existence and Uniqueness of the Determinant 728

A.19 Rigorous Proof of the Change of Variables Formula 732

A.20 Justifying Volume 0 739

A.21 Lebesgue Measure and Proofs for Lebesgue Integrals 741

A.22 Justifying the Change of Parametrization 759

A.23 Computing the Exterior Derivative 762

A.24 The Pullback 766

A.25 Proof of Stokes' Theorem 771

Appendix B Programs 783

B.1 Matlab Newton Program 783

B.2 Monte Carlo Program 784

B.3 Determinant Program 786.

Notes:

Includes bibliographical references (pages 789-790) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Classes of 1883 and 1884 Fund.

ISBN:

0130414085

OCLC:

47644373