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Vector Calculus, 2/E
Thomas H. Barr, Rhodes College

ISBN-10: 0130880051
ISBN-13: 9780130880055

Publisher: Prentice Hall
Copyright: 2001
Format: Cloth; 429 pp
Published: 11/07/2000

Suggested retail price: $129.20
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For one semester, sophomore-level courses in Vector Calculus and Multivariable Calculus.

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 approach, it is written with an assumption that the student may have computing facilities for two- and three-dimensional graphics, and for doing symbolic algebra.

  • NEW - Thorough coverage of constrained optimization/Lagrange multipliers—Along with second derivative tests.

    • Determines the nature of constrained local extrema. Ex.___

  • NEW - Student-tested laboratory and writing exercises.

    • Helps students investigate mathematical problems using software tools, and encourages them to practice their writing skills through experiences in the laboratory. Ex.___

  • NEW - More than 200 new exercises—Includes drills, applications, proofs, and “technologically active” projects.

    • Challenges students to use computer graphics and symbolic algebra in ways that enhance their understanding. Ex.___

  • NEW - Many new examples—Now total approximately 300

    • Many include new figures to aid students in visualization, others include new applications.

  • Emphasis on parameterization.

    • Encourages students to visualize with the aid of hand drawings and computers, in order to enhance their geometric intuition. Ex.___

  • Presentation/use of linear algebra as a tool.

    • Provides students with many conceptual and formal parallels with single-variable calculus. Ex.___

  • Early introduction to geometry in three-dimensional space—Along with Cylindrical and Spherical coordinates.

    • Prepares students for their later use in connection with the Chain Rule, and change of variables in double and triple integrals. Ex.___

  • Early introduction to matrix notation and the rudiments of linear algebra.

    • Familiarizes students with these topics so that it is easier to understand and build upon later material throughout the text. Ex.___

  • Technology oriented exercises and projects.

    • Gives students the opportunity to practice what they have learned, including using computers to produce a solution. Ex.___

  • Application motivated definitions.

    • Supplies students with a connection between “theory” and “applications.” Ex.___

  • Thorough coverage of constrained optimization/Lagrange multipliers—Along with second derivative tests.

    • Determines the nature of constrained local extrema. Ex.___

  • Student-tested laboratory and writing exercises.

    • Helps students investigate mathematical problems using software tools, and encourages them to practice their writing skills through experiences in the laboratory. Ex.___

  • More than 200 new exercises—Includes drills, applications, proofs, and “technologically active” projects.

    • Challenges students to use computer graphics and symbolic algebra in ways that enhance their understanding. Ex.___

  • Many new examples—Now total approximately 300

    • Many include new figures to aid students in visualization, others include new applications.



1. Coordinate and Vector Geometry.

Rectangular Coordinates and Distance. Graphs of Functions of Two Variables. Quadric Surfaces. Cylindrical and Spherical Coordinates. Vectors in Three-Dimensional Space. The Dot Product, Projection, and Work. The Cross Product and Determinants. Planes and Lines in R3. Vector-Valued Functions. Derivatives and Motion.



2. Geometry and Linear Algebra in Rn.

Vectors and Coordinate Geometry in Rn. Matrices. Linear Transformations. Geometry of Linear Transformations. Quadratic Forms.



3. Differentiation.

Graphs, Level Sets, and Vector Fields: Geometry. Limits and Continuity. Open Sets, Closed Sets, and Continuity. Partial Derivatives. Differentiation and the Total Derivative. The Chain Rule.



4. Applications of Differentiation.

The Gradient and Directional Derivative. Divergence and Curl. Taylor's Theorem. Local Extrema. Constrained Optimization and Lagrange Multipliers.



5. Integration.

Paths and Arclength. 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.



6. Fundamental Theorems.

The Fundamental Theorem for Path Integrals. Green's Theorem. The Divergence Theorem. Stokes's Theorem.



7. Laboratory Writing Projects.

Plotting Parameterized Surfaces. Making a Movie. A Mechanical Linkage. The Frenet Frame. Bézier Curves. Filling a Lake. Calculating Volume by Changing Coordinates. Predicting Eclipses.



Bibliography.


Answers to Selected Exercises.


Index.

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