Our overarching aim in writing this book is to build a bridge to enable engineers to better traverse the domains of the mathematical and physical worlds. Our focus is on neither the nuances of pure mathematics nor the phenomenology of physical devices but instead is on the mathematical tools used today in many engineering environments. We often compromise strict formalism for the sake of efficient exposition of mathematical tools. Whereas some results are fully derived, others are simply asserted, especially when detailed proofs would significantly lengthen the presentation. Thus, the book emphasizes method and technique over rigor and completeness; readers who require more of the latter can and should turn to many of the foundational works cited in the extensive bibliography.

Our specific objective is to survey topics in engineering-relevant applied mathematics, including multivariable calculus, vectors and tensors, ordinary differential equations, approximation methods, linear analysis, linear algebra, linear integral equations, and nonlinear dynamical systems. In short, the text fully explores linear systems and considers some effects of nonlinearity, especially those types that can be treated analytically. Many topics have geometric interpretations, identified throughout the book. Particular attention is paid to the notion of approximation via projection of an entity from a high-or even infinite-dimensional space onto a space of lower dimension. Another goal is to give the student the mathematical background to delve into topics such as dynamics, differential geometry, continuum mechanics, and computational methods; although the material presented is relevant to those fields, specific physical applications are mainly confined to some of the exercises. A final goal is to introduce the engineer to some of the notation and rigor of mathematics in a way used in many upper-level graduate engineering and applied mathematics courses.

This book is intended for use in a beginning graduate course in applied mathematics taught to engineers. It arose from a set of notes for such a course taught by the authors for more than 20 years in the Department of Aerospace and Mechanical Engineering at the University of Notre Dame.