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
Copyright 1999, 687 pp.
Sign up for future
mailings on this subject.
See other books about:
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,
Presents more difficult and longer proofs (e.g., proofs
of the Kantorovitch theorem, the implicit function theorem) in
- Includes a chapter on differential forms
written to be compatible with the more usual vector calculus
- Includes a section on numerical methods of integration
usually lacking in texts at this level.
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.
- 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.
Presentation of manifolds much easier than the standard
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.
Begins most chapters with a treatment of a linear problem
and then shows how the methods apply to corresponding
- Emphasizes computationally effective algorithms and proves
theorems by showing that those algorithms really work.
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
Contains abundant student-friendly aids:
Features a full set of examples and exercises:
- 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.
- 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).
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
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.