Vector Calculus, 1/e
Tom Barr, Rhodes College
Published October, 1996 by Prentice Hall Engineering/Science/Mathematics
Copyright 1997, 449 pp.
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This brief book presents an accessible treatment of multivariable calculus with an early emphasis on linear algebra as a tool. The organization of the text draws strong analogies with the basic ideas of elementary calculus (derivative, integral, and fundamental theorem). Traditional in its approach, it is written with an assumption that the reader may have computing facilities for two- and three-dimensional graphics and for doing symbolic algebra.
Emphasizes geometric intuition and visualization. The text
contains hundreds of figures and through the exposition and exercises, the
reader is encouraged to visualize with the aid of hand drawings and
Introduces geometry in three dimensional space early in the
book along with Cylindrical and Spherical coordinates, anticipating their
later use in connection with the Chain Rule and change of variables in
double and triple integrals.
Introduces matrix notation and the rudiments of linear algebra
early in the book; this material is then used to facilitate the exposition throughout the
rest of the book.
Includes approximately 1200 exercises that include drills,
applications, proofs, and "technologically active" projects.
Provides exercises and projects which are oriented toward the
use of technology. These are marked to indicate that a computer may be
helpful in producing a solution.
Many of the definitions throughout the book are motivated by
applications. In this way, "theory" and "applications" are joined.
1. Coordinate and Vector Geometry.
Rectangular Coordinates and Distance. Surfaces and Equations. Cylindrical and Spherical Coordinates. Vectors in 3-Dimensional Space. The Dot Product, Projection, and Work. The Cross Product and Determinants. Planes and Lines in Rgif/super_k.gif3. Ve
ctor-Valued Functions. Derivatives and Motion. Projects and Problems.
2. Geometry and Linear Algebra in Rgif/super_k.gifn.
Vectors and Coordinate Geometry in Rgif/super_k.gifn. Matrices. Linear Transformations. Geometry of Linear Transformation. Quadratic Forms.
Graphs, Level Sets, and Vector Fields: Geometry. Continuity. Open and Closed Sets and Continuity. Partial Derivatives. Differentiability and the Total Derivative. The Chain Rule. The Gradient and Directional Derivative. Divergence and Curl.
Mean Value Theorems: Taylor's Theorem. Local Extrema.
Paths, Curves, and Length, Line Integrals. Double Integrals. Triple Integrals. Parametrized Surfaces and Surface Area. Surface Integrals. Change of Variables in Double Integrals. Change of Variables in Triple Integrals.
5. Fundamental Theorems and Applications.
Path Independent Vector Fields. Green's Theorem. The Divergence Theorem. Stokes' Theorem.
Answers to Selected Exercises.