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Foundations of Applied Mathematics, Volume 1: Mathematical Analysis

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 Product Description

by Jeffrey Humpherys, Tyler J. Jarvis, and Emily J. Evans


2017 / xx + 689 pages / Hardcover / ISBN 978-1-611974-89-8 / List Price $89.00 / SIAM Member Price $62.30 / Order Code: OT152

Keywords: numerical linear algebra, resolvent methods, analysis, matrix theory, integration


This book provides the essential foundations of both linear and nonlinear analysis necessary for understanding and working in twenty-first century applied and computational mathematics. In addition to the standard topics, this text includes several key concepts of modern applied mathematical analysis that should be, but are not typically, included in advanced undergraduate and beginning graduate mathematics curricula. This material is the introductory foundation upon which algorithm analysis, optimization, probability, statistics, differential equations, machine learning, and control theory are built. When used in concert with the free supplemental lab materials, this text teaches students both the theory and the computational practice of modern mathematical analysis.

Foundations of Applied Mathematics, Volume 1: Mathematical Analysis includes several key topics not usually treated in courses at this level, such as uniform contraction mappings, the continuous linear extension theorem, Daniell–Lebesgue integration, resolvents, spectral resolution theory, and pseudospectra. Ideas are developed in a mathematically rigorous way and students are provided with powerful tools and beautiful ideas that yield a number of nice proofs, all of which contribute to a deep understanding of advanced analysis and linear algebra. Carefully thought out exercises and examples are built on each other to reinforce and retain concepts and ideas and to achieve greater depth. Associated lab materials are available that expose students to applications and numerical computation and reinforce the theoretical ideas taught in the text. The text and labs combine to make students technically proficient and to answer the age-old question, "When am I going to use this?"

This textbook is appropriate for advanced undergraduate or beginning graduate students in mathematics and, potentially, graduate students in physics, engineering, statistics, or computer science.

About the Author
Jeffrey Humpherys is a professor of mathematics at Brigham Young University, former Vice Chair of the SIAM Activity Group on Applied Mathematics Education, and a two-term member of the SIAM Education Committee. He is the recipient of a National Science Foundation CAREER award. His research spans a wide range of topics in applied and computational mathematics, from nonlinear partial differential equations to network sciences to machine learning.

Tyler Jarvis is a professor of mathematics at Brigham Young University whose research has primarily been in geometric problems arising from physics. He is the recipient of a National Science Foundation CAREER award and the Mathematical Association of America's Deborah and Franklin Tepper Haimo Award for Distinguished University Teaching of Mathematics.

Emily Evans is an assistant professor in the Department of Mathematics at Brigham Young University. Prior to earning her Ph.D. she spent seven years in industry as a software engineer. Her research interests include finite elements for domains with fractal boundaries, biological modeling, computational mechanics, the mathematics of computer animation, and network science.

ISBN 9781611974898

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 Product Reviews

  1. Comprehensive and readable analysis textbook: Perhaps the future of applied mathematics

    Posted by Hayden Ringer on 30th Nov 2017

    I did my undergraduate math degree at BYU under the three authors of this textbook. This book was in progress while I was in the program, along with the other three volumes. I owe a great deal to this textbook and the program built around it.

    The treatment of analysis in Volume 1 is essentially comprehensive at the upper undergraduate/master's degree level. It covers both linear and nonlinear analysis in their most general forms. For example, the treatment of linear analysis does not restrict itself to finite dimensional spaces, but explicitly presents the core theorems in a Banach space of arbitrary dimension. This approach continues into nonlinear analysis, where the Frechet derivative on an arbitrary Banach space is the fundamental object. As a result, students are prepared to explore the general theory of ordinary and partial differential equations, as well as the Calculus of Variations and optimal control theory (presented in the fourth volume of the series).

    While the adjective "applied" is part of the title of the book, it absolutely does not hand wave or avoid rigor. The proofs in the text are complete, and the exercises are intended to be very challenging. Many of them , if not most, ask students to prove fundamental results that are used later in the text. This gives the student a sense of accomplishment and helps them to think and understand the material at a deeper level.

    The associated Python labs complete the circle. The book provides the foundational theory, the exercises extend the understanding, and the labs give the practical application. For instance, in Chapter 3 (Inner Product Spaces), students learn about the general Gram-Schmidt process, and then in the associated lab, students code up the QR decomposition using modified Gram-Schmidt. QR continues to come up in later chapters and labs as an important tool in numerical linear algebra.

    In terms of style, the book is very readable, despite the density of content. Each chapter begins with a discussion of the questions that the soon-to-be-learned math will answer. The format is very clear: definitions and theorems are given clear bold headings (and descriptive names whenever possible), examples, applications, helpful notes, "unexamples", and "vistas" are in color-coded boxes. The organizational structure of the book makes it just as valuable as a reference text as it is for learning the material the first time.

    Since the book is published through SIAM, it is amazingly cheap, given that it has 709 pages. Professors can rest a little easier, knowing they aren't making their students scam themselves by paying hundreds of dollars for a paperback pamphlet.

    All-in-all, I cannot imagine a better or more complete textbook for analysis. Humpherys, Jarvis, and Evans have produced a fantastic work that could certainly become the standard text for this topic at the early graduate level.

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