高等工程數學

譚建國教授

國立成功大學土木工程學系

成大出版社

2023

ISBN 978-986-5635-92-3

摘要

    良好的數學修養對科技研究者之重要性如同充沛的體力對運動員一樣，不可或缺。工程數學為工學院的基礎科目，坊間不乏經典的工程數學外文書籍，但少有優良的中文教本。本書是根據作者歷年來講授高等工程數學課程的講義綜理成章，編撰而成，旨在提供一本適合大學部高年級生與研究生簡明嚴謹而深入淺出的中文課本。

    全書共六章：1. 多變數函數與向量微積分概要 2. 曲線座標 3. 傅立葉級數與傅立葉變換 4. 偏微分方程式與數理問題之解析 5. 複變函數理論與應用 6. 變分法與應用，取材以解析工程物理問題的偏微分方程式數學模式、複變函數理論以及變分法與應用為主，前三章為有關的數學預備知識。本書著重闡明基本概念與延伸的理論，以及解析數理問題的思路與方法，兼顧理論與應用、章節的連貫性、起承轉合與前後呼應。

    本書設定讀者具備解析常微分方程式、矩陣運算與向量分析等大學部基本工程數學基礎，必要的數學知識在各章節適時補充闡明，習題多為輔助讀者理解內文的問題與註解，以期溫故知新，有助融會貫通，並啟發研習的興趣。

An Introduction to Advanced Engineering Mathematics

Tarn, Jiann-Quo  

Abstract

    Mathematical capability is essential to researchers as physical strength to athletes. As such, Engineering Mathematics is a required course in engineering school. There is no lack of good monographs on its theory and application. However, relevant textbooks in Chinese that are concise yet provide the reader with sufficient details and rigor are in short supply. This book is essentially based on author’s lecture notes on Advanced Engineering Mathematics given over many years. The purpose is to present treatments of a number of important topics that are useful in many fields of application for advanced undergraduates and graduate students in engineering.

The book consists of six chapters: 1. Higher-Dimensional Calculus and Essential Vector Analysis. 2. Curvilinear Coordinates. 3. Fourier Series and Fourier Transforms. 4. Partial Differential Equations and Applications to Problems of Mathematical Physics. 5. Theory of Complex Variables and Applications. 6. Calculus of Variations and Applications. The first three chapters are prerequisites of the materials that follow. Throughout the book, basic idea and solution approach are emphasized, with many examples to illustrate the underlying theory and solution methods. Exercise problems that complement the topics covered are included in sections of each chapter.

A reasonable command of topics in elementary courses of engineering mathematics is presumed, such as ordinary differential equations, matrices and linear algebra, and vector analysis. For easy understanding of the contents, background knowledge is provided, and essential prerequisites supplemented wherever appropriate. The book is self-contained and serves as a suitable text for the course of Advanced Engineering Mathematics.