This text is intended for a one-semester course in the calculus of functions of several variables and vector analysis, which is normally taught at the sophomore level. In addition to making changes and improvements throughout the text, we have also attempted to convey a sense of excitement, relevance, and importance of the subject matter.

Sometimes courses in vector calculus are preceded by a first course in linear algebra, but this is not an essential prerequisite. We require only the bare rudiments of matrix algebra, and the necessary concepts are developed in the text. If this course is preceded by a course in linear algebra, the instructor will have no difficulty enhancing the material. However, we do assume a knowledge of the fundamentals of one-variable calculus—the process of differentiation and integration and their geometric and physical meaning as well as a knowledge of the standard functions, such as the trigonometric and exponential functions.

The text includes much of the basic theory as well as many concrete examples and problems. Some of the technical proofs for theorems in Chapters 2 and 5 are given in optional sections that are readily available on the Book Companion Web Site at www.whfreeman.com/marsdenvc6e (see the description on the next page). Section 2.2, on limits and continuity, is designed to be treated lightly and is deliberately brief. More sophisticated theoretical topics, such as compactness and delicate proofs in integration theory, have been omitted, because they usually belong to a more advanced course in real analysis.

Computational skills and intuitive understanding are important at this level, and we have tried to meet this need by making the book concrete and student-oriented. For example, although we formulate the definition of the derivative correctly, it is done by using matrices of partial derivatives rather than abstract linear transformations. We also include a number of physical illustrations such as fluid mechanics, gravitation, and electromagnetic theory, and from economics as well, although knowledge of these subjects is not assumed.

A special feature of the text is the early introduction of vector fields, divergence, and curl in Chapter 4, before integration. Vector analysis often suffers in a course of this type, and the present arrangement is designed to offset this tendency. To go even further, one might consider teaching Chapter 3 (Taylor’s theorems, maxima and minima, Lagrange multipliers) after Chapter 8 (the integral theorems of vector analysis).

This sixth edition was completely redesigned, but retains and improves on the balance between theory, applications, optional material, and historical notes that was present in earlier editions.

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We are excited about this new edition of Vector Calculus, especially the inclusion of many new exercises and examples. The exercises have been graded from less difficult to more difficult, allowing instructors to have more flexibility in assigning practice problems. The modern redesign emphasizes the pedagogical features, making the text more concise, student-friendly, and accessible. The quality of the art work has been significantly improved, especially for the crucial three-dimensional figures, to better reflect key concepts to students. We have also trimmed some of the historical material, making it more relevant to the mathematics under discussion. Finally, we have moved some of the more difficult discussions in the fifth edition—such as those on Conservation Laws, the derivation of Euler’s Equation of a Perfect Fluid, and a discussion of the Heat Equation—to the Book Companion Web Site. We hope that the reader will be equally pleased.

The following electronic and print supplements are available with Vector Calculus, Sixth Edition:

Jerry Marsden and Tony Tromba,

Caltech and UC Santa Cruz, Summer 2010.

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