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Vector Calculus

Jerrold E. Marsden and Anthony Tromba
Publisher: 
W. H. Freeman
Publication Date: 
2003
Number of Pages: 
704
Format: 
Hardcover
Edition: 
5
Price: 
108.95
ISBN: 
0716749920
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
William J. Satzer
, on
12/11/2012
]

This is the sixth edition of a popular advanced calculus text. It has been completely redesigned, yet retains the basic structure and approach of the earlier editions. Part of the appeal of the book has been the balance of its treatment across theory, applications, historical notes and optional material. That remains. What has been added are new examples, more exercises (with a clearer grading of difficulty) and improved artwork with significantly improved three-dimensional figures.

One of the reasons this book has been as successful is probably its focus on the average student. The authors have taken pains throughout to walk students through detailed explanations and worked-out examples, and then test their understanding with more routine exercises before presenting more challenging problems. Their object is simply to take a direct, fairly concrete path to teaching multivariable calculus without many detours. So, for example, instead of introducing the more abstract language of linear transformations the authors describe the derivative in terms of a matrix of partial derivatives.

The topics follow a traditional path, from the geometry of Euclidean space and the rudiments of matrix algebra to differentiation and integration in two and three dimensions. The book concludes with the theorems of Green, Gauss and Stokes, first presented in more traditional form and then in the language of differential forms. The authors handle basic theory cautiously. The treatment of limits and continuity is short and sweet, and more complicated proofs are made available at a companion web site. More sophisticated questions involving things like compactness or conditions for integrability are generally deferred to a later course.

The companion web site provides extensive “Additional Material”, almost a shadow textbook of more than a hundred pages, with theorems and proofs, more examples, historical material and advanced applications. This is a wonderful source of enrichment material. Besides providing rigorous proofs behind material in the main text, the additional material also provides supplementary exercises and some terrific excursions: the Korteweg-deVries equation, Principle of Least Action, the Sunshine formula (finding the position of the sun as a function of latitude and day of the year), and more.

The presentation, both in the main text and the supplementary material, is consistently attractive and the ideas are engaging. It offers the instructor a good deal of flexibility in tailoring a course to deal with a mix of student backgrounds and abilities.


Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

 1. THE GEOMETRY OF EUCLIDEAN SPACE
    1.1 Vectors in Two- and Three-Dimensional Space
    1.2 The Inner Product, Length, and Distance
    1.3 Matrices, Determinants, and the Cross Product
    1.4 Cylindrical and Spherical Coordinates
    1.5 n-Dimensional Euclidean Space
    
2. DIFFERENTIATION SPACE
    2.1 The Geometry of Real-Valued Functions
    2.2 Limits and Continuity
    2.3 Differentiation
    2.4 Introduction to Paths
    2.5 Properties of the Derivative
    2.6 Gradients and Directional Derivatives
    
3. HIGHER-ORDER DERIVATIVES: MAXIMA AND MINIMA
    3.1 Iterated Partial Derivatives
    3.2 Taylor's Theorem
    3.3 Extrema of Real-Valued Functions
    3.4 Constrained Extrema and Lagrange Multipliers
    3.5 The Implicit Function Theorem
    
4. VECTOR-VALUED FUNCTIONS
    4.1 Acceleration and Newton's Second Law
    4.2 Arc Length
    4.3 Vector Fields
    4.4 Divergence and Curl
    
5. DOUBLE AND TRIPLE INTEGRALS
    5.1 Introduction
    5.2 The Double Integral Over a Rectangle
    5.3 The Double Integral Over More General Regions
    5.4 Changing the Order of Integration
    5.5 The Triple Integral
    
6. THE CHANGE OF VARIABLES FORMULA AND APPLICATIONS OF INTEGRATION
    6.1 The Geometry of Maps from R2 to R2
    6.2 The Change of Variables Theorem
    6.3 Applications of Double and Triple
    6.4 Improper Integrals
    
7. INTEGRALS OVER PATHS AND SURFACES
    7.1 The Path Integral
    7.2 Line Integrals
    7.3 Parametrized Surfaces
    7.4 Area of a Surface
    7.5 Integrals of Scalar Functions Over Surfaces
    7.6 Surface Integrals of Vector Functions
    7.7 Applications to Differential Geometry, Physics and Forms of Life
    
8. THE INTEGRAL THEOREMS OF VECTOR ANALYSIS
    8.1 Green's Theorem
    8.2 Stokes' Theorem
    8.3 Conservative Fields
    8.4 Gauss' Theorem
    8.5 Applications to Physics, Engineering, and Differential Equations
    8.6 Differential Forms
    
     
    

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