Vector Calculus

by Colley, Susan J.

• ISBN13: 9780131858749
• ISBN10: 0131858742
• Edition: 3rd
• Format: Hardcover
• Copyright: 2005-03-16
• Publisher: Pearson

Summary

This text uses the language and notation of vectors and matrices to clarify issues in multivariable calculus. Accessible to anyone with a good background in single-variable calculus, it presents more linear algebra than usually found in a multivariable calculus book. Colley balances this with very clear and expansive exposition, many figures, and numerous, wide-ranging exercises. Instructors will appreciate Colleyrs"s writing style, mathematical precision, level of rigor, and full selection of topics treated. Vectors:Vectors in Two and Three Dimensions. More About Vectors. The Dot Product. The Cross Product. Equations for Planes; Distance Problems. Somen-Dimensional Geometry. New Coordinate Systems.Differentiation in Several Variables:Functions of Several Variables; Graphing Surfaces. Limits. The Derivative. Properties; Higher-Order Partial Derivatives; Newtonrs"s Method. The Chain Rule. Directional Derivatives and the Gradient.Vector-Valued Functions:Parametrized Curves and Kepler's Laws. Arclength and Differential Geometry. Vector Fields: An Introduction. Gradient, Divergence, Curl, and the Del Operator.Maxima and Minima in Several Variables:Differentials and Taylor's Theorem. Extrema of Functions. Lagrange Multipliers. Some Applications of Extrema.Multiple Integration:Introduction: Areas and Volumes. Double Integrals. Changing the Order of Integration. Triple Integrals. Change of Variables. Applications of Integration.Line Integrals:Scalar and Vector Line Integrals. Green's Theorem. Conservative Vector Fields.Surface Integrals and Vector Analysis:Parametrized Surfaces. Surface Integrals. Stokes's and Gauss's Theorems. Further Vector Analysis; Maxwell's Equations.Vector Analysis in Higher Dimensions:An Introduction to Differential Forms. Manifolds and Integrals ofk-forms. The Generalized Stokes's Theorem. For all readers interested in multivariable calculus.

Author Biography

Susan Colley is the Andrew and Pauline Delaney Professor of Mathematics at Oberlin College and currently Chair of the Department, having also previously served as Chair. She received S.B. and Ph.D. degrees in mathematics from the Massachusetts Institute of Technology prior to joining the faculty at Oberlin in 1983. Her research focuses on enumerative problems in algebraic geometry, particularly concerning multiple-point singularities and higher-order contact of plane curves. Professor Colley has published papers on algebraic geometry and commutative algebra, as well as articles on other mathematical subjects. She has lectured internationally on her research and has taught a wide range of subjects in undergraduate mathematics. Professor Colley is a member of several professional and honorary societies, including the American Mathematical Society, the Mathematical Association of America, Phi Beta Kappa, and Sigma Xi.

Table of Contents

Preface	ix
To the Student: Some Preliminary Notation	xiii
Vectors	1	(78)
Vectors in Two and Three Dimensions	1	(7)
More About Vectors	8	(10)
The Dot Product	18	(9)
The Cross Product	27	(12)
Equations for Planes; Distance Problems	39	(8)
Some n-dimensional Geometry	47	(14)
New Coordinate Systems	61	(11)
True/False Exercises for Chapter 1	72	(1)
Miscellaneous Exercises for Chapter 1	73	(6)
Differentiation in Several Variables	79	(98)
Functions of Several Variables; Graphing Surfaces	79	(14)
Limits	93	(15)
The Derivative	108	(17)
Properties; Higher-order Partial Derivatives; Newton's Method	125	(14)
The Chain Rule	139	(14)
Directional Derivatives and the Gradient	153	(17)
True/False Exercises for Chapter 2	170	(1)
Miscellaneous Exercises for Chapter 2	171	(6)
Vector-Valued Functions	177	(53)
Parametrized Curves and Kepler's Laws	177	(13)
Arclength and Differential Geometry	190	(18)
Vector Fields: An Introduction	208	(6)
Gradient, Divergence, Curl, and the Del Operator	214	(9)
True/False Exercises for Chapter 3	223	(1)
Miscellaneous Exercises for Chapter 3	223	(7)
Maxima and Minima in Several Variables	230	(59)
Differentials and Taylor's Theorem	230	(15)
Extrema of Functions	245	(14)
Lagrange Multipliers	259	(14)
Some Applications of Extrema	273	(11)
True/False Exercises for Chapter 4	284	(1)
Miscellaneous Exercises for Chapter 4	285	(4)
Multiple Integration	289	(74)
Introduction: Areas and Volumes	289	(3)
Double Integrals	292	(17)
Changing the Order of Integration	309	(3)
Triple Integrals	312	(10)
Change of Variables	322	(21)
Applications of Integration	343	(14)
True/False Exercises for Chapter 5	357	(1)
Miscellaneous Exercises for Chapter 5	358	(5)
Line Integrals	363	(42)
Scalar and Vector Line Integrals	363	(18)
Green's Theorem	381	(9)
Conservative Vector Fields	390	(10)
True/False Exercises for Chapter 6	400	(1)
Miscellaneous Exercises for Chapter 6	401	(4)
Surface Integrals and Vector Analysis	405	(72)
Parametrized Surfaces	405	(14)
Surface Integrals	419	(20)
Stokes's and Gauss's Theorems	439	(18)
Further Vector Analysis; Maxwell's Equations	457	(12)
True/False Exercises for Chapter 7	469	(1)
Miscellaneous Exercises for Chapter 7	470	(7)
Vector Analysis in Higher Dimensions	477	(32)
An Introduction to Differential Forms	477	(6)
Manifolds and Integrals of k-forms	483	(15)
The Generalized Stokes's Theorem	498	(8)
True/False Exercises for Chapter 8	506	(1)
Miscellaneous Exercises for Chapter 8	507	(2)
Suggestions for Further Reading	509	(2)
Answers to Selected Exercises	511	(32)
Index	543