** ** This book explores the standard problem-solving techniques of multivariable mathematics — integrating vector algebra ideas with multivariable calculus and differential equations. ** ** Unique coverage including, the introduction of vector geometry and matrix algrebra, the early introduction of the gradient vector as the key to differentiability, optional numerical methods. ** ** For any reader interested in learning more about this discipline.

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This text explores the standard problem-solving techniques of multivariable mathematics -- integrating vector algebra ideas with multivariable calculus and differential equations.

This book covers material that is often studied after a first course in one-variable calculus, namely the algebra and geometry of vectors and matrices, multivariable and vector calculus, and differential equations, including systems. The branches of these three areas are strongly intertwined and we've designed our treatment to display the connections effectively. Our aim has been to teach basic problem solving, both pure and applied, in a framework that is mathematically coherent, while allowing for selective emphasis on traditional rigor.

While the sequence of topics follows rather traditional lines of mathematical classification, the actual route taken may vary widely from course to course. An underlying theme is the encouragement of geometric thinking in two and three dimensions, extended to arbitrary dimension when it's useful to do so. Thus, most of Chapters 1 and 2 on vectors and matrices is prerequisite for the rest of the book, but otherwise there is considerable flexibility for course scheduling. Chapter 3 on linear algebra, with an introduction to general vector spaces and linear transformations, is included for those who want to cover this material at some point, but none of it is prerequisite for later chapters. In particular the material on differentiability in Chapter 5 is organized so that the motivation for the definition depends on gradient vectors rather than linear transformations.

For this edition the exposition has been completely rewritten in many places and, in addition to Chapter 3, a number of topics that are optional additions to a basic course have been added, as follows:

Additional emphasis on scientific applications in Section 113 of Chapter 2.

Subsection on vector integrals in Chapter 4, Section 1.

Subsections on quadric surfaces in Chapter 4.

Subsection on flow lines in Chapter 6, Section 1.

Subsection on use of the chain rule in coordinate changes.

Expanded treatment of the second-derivative criterion for extrema.

Section 5 on centroids and moments in Chapter 7.

Section 6 on application of improper integrals in Chapter 7.

Section 4, Chapter 8 relating flow lines, divergence, and curl.

Subsection on finding potentials in Chapter 9, Section 2.

Additional subsection on flows in Chapter 12, Section 1.

More efficient computation of exponential matrices in Chapter 13, Section 2.

Chapter on infinite series, with applications to differential equations.

Sections in Chapter 4 on computer plotting of curves and surfaces.

Subsection on the steepest ascent method.

Subsection on the midpoint and Simpson rules for multiple integrals.

Subsection on Newton's method for vector functions.

Java applets for the graphical and numerical work.

The figures are a salient feature of the text, including those in the answer section and the one on the cover, which represents a trajectory of a Lotka-Volterra system modified for three species and discussed in Exercise 34 in Chapter 12, Section 4.

The impetus for this edition came from George Lobell, whose knowledge of the field and continued support has helped us a great deal. Allan Gunter read the entire text, making insightful suggestions and working all the exercises; his collaboration was invaluable. Jeanne Audino's experienced and good humored oversight of the production has made working out final details a pleasure rather than a chore. Corrections can be sent to rewn@dartmouth.edu or hft@math.princeton.edu.

Richard E. Williamson

Hale F. Trotter

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