Vector Calculus, 2nd edition

• Thomas H. Barr

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Overview

This brief book presents an accessible treatment of multivariable calculus with an early emphasis on linear algebra as a tool. Its organization 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 reader may have computing facilities for two- and three-dimensional graphics, and for doing symbolic algebra. KEY TOPICS: Chapter topics include coordinate and vector geometry, differentiation, applications of differentiation, integration, and fundamental theorems. MARKET: For those with knowledge of introductory calculus in a wide range of disciplines including—but not limited to—mathematics, engineering, physics, chemistry, and economics.

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.