John H. Hubbard Knihy

Traditional courses on differential equations emphasize solution techniques, but most equations, especially nonlinear ones in higher dimensions, lack elementary solutions. A century ago, Poincaré revolutionized the field with his insights on the three-body problem, proposing that differential equations define families of parametric curves in higher dimensions, shifting the focus to understanding their geometry and behavior. The first part of this series addresses differential equations in one dimension, where geometric complications are limited, and aims to extend these methods to higher dimensions. This transition presents greater challenges, with some chapters being more advanced than Part I, yet still accessible to undergraduates. Poincaré's groundbreaking ideas took time for the mathematical community to fully appreciate, and prior to computer graphics, the notion of teaching this material at the undergraduate level seemed implausible. However, advancements now allow us to incorporate Poincaré's approach into the curriculum, enabling experimental exploration of these concepts. The companion software, MacMath, along with its extensions, is designed to animate these ideas, enhancing the learning experience.

This corrected third printing retains the authors'main emphasis on ordinary differential equations. It is most appropriate for upper level undergraduate and graduate students in the fields of mathematics, engineering, and applied mathematics, as well as the life sciences, physics and economics. The authors have taken the view that a differential equations theory defines functions; the object of the theory is to understand the behaviour of these functions. The tools the authors use include qualitative and numerical methods besides the traditional analytic methods, and the companion software, MacMath, is designed to bring these notions to life.

This text covers most of the standard topics in multivariate calculus and part of a standard first course in linear algebra. It focuses on underlying ideas, integrates theory and applications, offers a host of pedagogical aids, and features coverage of differential forms and an emphasis on numerical methods to prepare students for modern applications of mathematics. *Covers important material that is usually omitted. *Presents more difficult and longer proofs (e.g. Proofs of the Kantorovitch theorem, the implicit function theorem) in an appendix. *Makes a careful distinction between vectors and points. *Features an innovative approach to the implicit function theorem and inverse function theorem using Newton's method. *Always emphasizes the underlying meaning - what is really going on (generally, with a geometric interpretation) - eg. The chain rule is a composition of linear transformations; the point of the implicit function theorem is to guarantee that under certain circumstances, non-linear equations have solutions. *Integrates theory and applications. *Begins most chapters with a treatment of a linear problem and then shows how the 7 methods apply to corresponding non-linear p