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Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, 2/E
John H. Hubbard, Cornell University
Barbara Burke Hubbard

ISBN-10: 0130414085
ISBN-13: 9780130414083

Publisher: Pearson
Copyright: 2002
Format: Cloth; 668 pp
Published: 09/07/2001
Status: Out of Print

Net price: $90.00 (What's this?)

We're sorry, this product is no longer available.
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For an undergraduate course in Vector or Multivariable Calculus for math, engineering, and science majors.

Using a dual presentation that is rigorous and comprehensive—yet exceptionally student-friendly in approach—this text covers most of the standard topics in multivariate calculus and part of a standard first course in linear algebra. It focuses in underlying ideas, integrates theory and applications, offers a host of pedagogical aids, features coverage of differential forms and emphasizes numerical methods to prepare students for modern applications of mathematics.

NEW - Revised and expanded content—Including new discussions of functions; complex numbers; Lebesgue integration; orientation; forms restricted to vector spaces; and expanded discussions of subsets and subspaces of Rⁿ; probability; and more.
- Makes text now a little easier and clearer.

NEW - Approximately 875 exercises—270 more exercises than the previous edition.
- Reinforces students' understanding of the material.

NEW - Approximately 294 examples—50 more examples than the previous edition.
- Deepens students' understanding of concepts and theorems by allowing them to see each step of the process.

NEW - 50 more figures and tables than the previous edition.
- Makes text a bit easier.

NEW - Student Solutions Manual.
- Gives students detailed solutions to several hundred exercises and provides a source of additional examples.

NEW - Pictures of mathematicians.

Unified approach to vector calculus, linear algebra, and differential forms.
- Gives students a better understanding of all three areas and shows how they all relate to each other.

Unique treatment of differential forms.
- Shows students how differential forms in three dimensions translate into the language of vector calculus.

Stresses computationally effective algorithms—Both for computations and for underlying theory.
- Illustrates for students how mathematics is done today.

More difficult and longer proofs in the appendix.
- Allows more advanced students to use the book at a higher level and beginning students to focus on statements and becoming at ease with techniques rather than being intimidated by the technical details of proofs.

Emphasis on the correspondence between different mathematical languages—Including algebra and geometry.
- Presents material as more intuitive and conceptual.

Integrates theory and application.
- Emphasizes computationally effective algorithms and proves theorems by showing that those algorithms really work.

Begins most chapters with a treatment of a linear problem.
- Shows students how the methods apply to “corresponding” non-linear problems.

Instant exercises with solutions in footnotes.
- Encourages students to read the text actively.

Revised and expanded content—Including new discussions of functions; complex numbers; Lebesgue integration; orientation; forms restricted to vector spaces; and expanded discussions of subsets and subspaces of Rⁿ; probability; and more.
- Makes text now a little easier and clearer.

Approximately 875 exercises—270 more exercises than the previous edition.
- Reinforces students' understanding of the material.

Approximately 294 examples—50 more examples than the previous edition.
- Deepens students' understanding of concepts and theorems by allowing them to see each step of the process.

50 more figures and tables than the previous edition.
- Makes text a bit easier.

Student Solutions Manual.
- Gives students detailed solutions to several hundred exercises and provides a source of additional examples.

Pictures of mathematicians.

(Note: Each chapter begins with an Introduction and ends with Review Exercises.)

0. Preliminaries.

Reading Mathematics. Quantifiers and Negation. Set Theory. Functions. Real Numbers. Infinite Sets. Complex Numbers.

1. Vectors, Matrices, and Derivatives.

Introducing the Actors: Points and Vectors. Introducing the Actors: Matrices. A Matrix as a Transformation. The Geometry of Rⁿ. Limits and Continuity. Four Big Theorems. Differential Calculus. Rules for Computing Derivatives. Mean Value Theorem and Criteria for Differentiability.

2. Solving Equations.

The Main Algorithm: Row Reduction. Solving Equations Using Row Reduction. Matrix Inverses and Elementary Matrices. Linear Combinations, Span, and Linear Independence. Kernels, Images, and the Dimension Formula. An Introduction to Abstract Vector Spaces. Newton's Method. Superconvergence. The Inverse and Implicit Function Theorems.

3. Higher Partial Derivatives, Quadratic Forms, and Manifolds.

Manifolds. Tangent Spaces. Taylor Polynomials in Several Variables. Rules for Computing Taylor Polynomials. Quadratic Forms. Classifying Critical Points of Functions. Constrained Critical Points and Lagrange Multipliers. Geometry of Curves and Surfaces.

4. Integration.

Defining the Integral. Probability and Centers of Gravity. What Functions Can Be Integrated? Integration and Measure Zero (Optional). Fubini's Theorem and Iterated Integrals. Numerical Methods of Integration. Other Pavings. Determinants. Volumes and Determinants. The Change of Variables Formula. Lebesgue Integrals.

5. Volumes of Manifolds.

Parallelograms and Their Volumes. Parameterizations. Computing Volumes of Manifolds. Fractals and Fractional Dimension.

6. Forms and Vector Calculus.

Forms on Rⁿ. Integrating Form Fields over Parameterized Domains. Orientation of Manifolds. Integrating Forms over Oriented Manifolds. Forms and Vector Calculus. Boundary Orientation. The Exterior Derivative. The Exterior Derivative in the Language of Vector Calculus. The Generalized Stokes's Theorem. The Integral Theorems of Vector Calculus. Potentials.

Appendix A: Some Harder Proofs.

Arithmetic of Real Numbers. Cubic and Quartic Equations. Two Extra Results in Topology. Proof of the Chain Rule. Proof of Kantorovich's Theorem. Proof of Lemma 2.8.5 (Superconvergence). Proof of Differentiability of the Inverse Function. Proof of the Implicit Function Theorem. Proof of Theorem 3.3.9: Equality of Crossed Partials. Proof of Proposition 3.3.19. Proof of Rules for Taylor Polynomials. Taylor's Theorem with Remainder. Proof of Theorem 3.5.3 (Completing Squares). Geometry of Curves and Surfaces: Proofs. Proof of the Central Limit Theorem. Proof of Fubini's Theorem. Justifying the Use of Other Pavings. Existence and Uniqueness of the Determinant. Rigorous Proof of the Change of Variables Formula. Justifying Volume 0. Lebesgue Measure and Proofs for Lebesgue Integrals. Justifying the Change of Parameterization. Computing the Exterior Derivative. The Pullback. Proof of Stokes' Theorem.

Appendix B.

MATLAB Newton Program. Monte Carlo Program. Determinant Program.

Bibliography.

Index.

Vector Calculus (Mathematics)
Multivariable Calculus (Mathematics)

Author Bios

John H. Hubbard (BA Harvard University, PhD University of Paris) is professor of mathematics at Cornell University and at the University of Provence in Marseilles he is the author of several books on differential equations. His research mainly concerns complex analysis, differential equations, and dynamical systems. He believes that mathematics research and teaching are activities that enrich each other and should not be separated.

Barbara Burke Hubbard (BA Harvard University) is the author of The World According to Wavelets, which was awarded the prix d'Alembert by the French Mathematical Society in 1996.

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