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Preface My Goals For This Book Science and engineering students depend heavily on concepts of mathematical modeling. In an age where almost everything is done on a computer, it is my conviction that students of engineering and science are better served if they understand and “own” the underlying mathematics that the computers are doing on their behalf. Mathematics is a necessary language for doing engineering and science. This will remain true no matter how good computation becomes. I repeatedly tell students that it is risky to accept computer calculations without having done some parallel closed-form modeling to benchmark the computer results. Without such benchmarking and validation, how do we know that the computer isn’t talking nonsense? Finally, I find it satisfying and fun to do mathematical manipulations that explain how or why something happens, and to use mathematics to obtain corresponding numerical data or predictions. Thus, as it was for the first edition, my primary goal for this second edition remains to engage the reader in developing a foundation for mathematical modeling. Further, knowing that mathematical models are built in a range of disciplines—including physics, biology, ecology, economics, sociology, military strategy, as well as all of the many branches of engineering—and knowing that mathematical modeling is comprised of a very diverse set of skills and tools, I focused on techniques of particular interest to engineers, scientists, and others who model continuous systems. xiii xiv Preface Features of This Edition Aided by a variety of reviewers’ comments and suggestions, this second edition features: • A more formal statement of a principled approach to mathematical modeling (in Chapter 1). Ten principles are articulated and invoked as applications are developed, and each of them is identified by a key word (see below). • Some 360 probl