## Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach, 1/e

John H. Hubbard, Cornell University
Barbara Burke Hubbard

Published September, 1998 by Prentice Hall Engineering/Science/Mathematics

Cloth
ISBN 0-13-657446-7

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Vector Calculus-Mathematics

Multivariable Calculus-Mathematics

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 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, e.g.:

• Includes a chapter on differential forms — written to be compatible with the more usual vector calculus approach.
• Includes a section on numerical methods of integration — usually lacking in texts at this level.
Presents more difficult and longer proofs (e.g., proofs of the Kantorovitch theorem, the implicit function theorem) in an appendix.
• This allows more advanced students to use the book at a higher level and beginning students to focus on understanding statements and becoming at ease with techniques rather than being intimidated by technical details of proofs.
Features a much more simplified approach to multiple integrals (i.e., the change of variables formula using general pavings) than that found in current literature.
Presentation of manifolds much easier than the standard approach.
Emphasizes the correspondences between different mathematical languages — e.g., algebra and geometry.
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) — e.g., 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 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 and then shows how the methods apply to “corresponding” non-linear problems.
Notation is consistent — e.g., vectors are written as column vectors to avoid ambiguity between a matrix or its transpose.
Encourages students to make use of computers where appropriate (row reduction, numerical methods of integration, visualizing parametrizations, etc.).
Contains abundant “student-friendly” aids:
• Provides frequent references to previous definitions, examples, etc. — making it easier for students to review needed background as they progress.
• Includes margin notes.
• Makes generous use of underbraces and overbraces to clarify difficult equations.
• Uses the same numbering system throughout the text for theorems, lemmas, propositions, corollaries, examples, equations, definitions, etc. to make it easier for instructor to refer to a specific part of the text.
Features a full set of examples and exercises:
• Offers approximately 200 examples.
• Features nearly 500 chapter-end exercises, grouped by section, that range from very easy to quite difficult.
• Presents occasional self-assessment “mini- exercises” within the text (with answers in footnotes).

0. Preliminaries.

Introduction. Reading Mathematics. How to Negate Mathematical Statements. Set Theory. Real Numbers. Infinite Sets and Russell's Paradox. Complex Numbers. Exercises for Chapter 0.

1. Vectors, Matrices, and Derivatives.

Introduction. Introducing the Actors: Vectors. Introducing the Actors: Matrices. A Matrix as a Transformation. The Geometry of R^n. Convergence and Limits. Four Big Theorems. Differential Calculus. Rules for Computing Derivatives. Criteria for Differentiability. Exercises for Chapter 1.

2. Solving Equations.

Introduction. The Main Algorithm: Row Reduction. Solving Equations Using Row Reduction. Matrix Inverses and Elementary Matrices. Linear Combinations, Span, and Linear Independence. Kernels and Images. Abstract Vector Spaces. Newton's Method. Superconvergence. The Inverse and Implicit Function Theorems. Exercises for Chapter 2.

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

Introduction. Curves and Surfaces. Manifolds. Taylor Polynomials in Several Variables. Rules for Computing Taylor Polynomials. Quadratic Forms. Classifying Critical Points. Constrained Critical Points and Lagrange Multipliers. Geometry of Curves and Surfaces. Exercises for Chapter 3.

4. Integration.

Introduction. Defining the Integral. Probability and Integrals. What Functions Can Be Integrated? Integrated 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. Improper Integrals. Exercises for Chapter 4.

5. Lengths of Curves, Areas of Surfaces,...

Introduction. Parallelograms and their Volumes. Arc Length. Surface Area. Volume of Manifolds. Fractals and Fractional Dimension. Exercises for Chapter 5.

6. Forms and Vector Calculus.

Introduction. Forms as Integrands over Oriented Domains. Forms on Rn. Integrating Form Fields over Parametrized Domains. Forms and Vector Calculus. Orientation and Integration of Form Fields. Boundary Orientation. The Exterior Derivative. The Exterior Derivative in the Language of Vector Calculus. Generalized Stokes' Theorem. The Integral Theorems of Vector Calculus. Potentials. Exercises for Chapter 6.

Appendix A. Some Harder Proofs.

Introduction. Proof of the Chain Rule. Proof of Kantorovitch's Theorem. Proof of Lemma 2.8.4 (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). Proof of Propositions 3.8.1 and 3.8.12 and 3.8.13 (Frenet Formulas). 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. A Few Extra Results in Topology. Proof of the Dominated Convergence Theorem. Justifying the Change of Parametrization. Proof of Theorem 6.7.3 (Computing the Exterior Derivative). The Pullback. Proof of Stokes' Theorem. Exercises.

Appendix B. Programs.

MATLAB Newton Program. Monte Carlo Program. Determinant Program.

Index.