Advanced Mathematics for Engineers and Physicists

This book is designed to be an introductory course to some basic chapters of Advanced Mathematics for Engineering and Physics students, researchers in different branches of Applied Mathematics and anyone wanting to improve their mathematical knowledge by a clear, live, self-contained and motivated text.
Here, one can find different topics, such as differential (first order or higher order) equations, systems of differential equations, Fourier series, Fourier and Laplace transforms, partial differential equations, some basic facts and applications of the calculus of variations and, last but not least, an original and more intuitive introduction to probability theory. All these topics are carefully introduced, with complete proofs, motivations, examples, applications, problems and exercises, which are completely solved at the end of the book.

We added a generous supplementary material (11.1) with a self-contained and complete introduction to normed, metric and Hilbert spaces. Since we used some topics from complex function theory, we also introduced in Chapter 11 a section (11.2) with the basic facts in this important field.

What a reader needs for a complete understanding of this book? For a deep understanding of this book, it is required to take a course in undergraduate calculus and linear algebra. We mostly tried to use the engineering intuition instead of insisting on mathematical tricks. The main feature of the material presented here is its clarity, motivation and the genuine desire of the authors to make extremely transparent the "mysterious" mathematical tools that are used to describe and organize the great variety of impressions that come to the searching mind, from the infinite complexity of Nature.

The book is recommended not only to engineering and physics students or researchers but also to junior students in mathematics because it shows the connection between pure mathematics and physical phenomena, which always supply motivations for mathematical discoveries.

這本書旨在為工程和物理學專業的學生以及應用數學不同領域的研究人員提供一門基礎高等數學的入門課程，同時也適合任何希望通過一本清晰、生動、自成一體且具有動機的教材來提高數學知識的人。

在這本書中，讀者可以找到不同的主題，如微分（一階或高階）方程、微分方程組、傅立葉級數、傅立葉和拉普拉斯變換、偏微分方程、變分計算的一些基本事實和應用，以及對概率論的原創且更直觀的介紹。所有這些主題都被仔細介紹，附有完整的證明、動機、例子、應用、問題和練習題，並且在書末完全解答。

我們還添加了一個豐富的補充材料（11.1），其中包含對規範、度量和希爾伯特空間的自成一體且完整的介紹。由於我們使用了一些復變函數理論的主題，因此在第11章中還引入了一個部分（11.2），介紹了這一重要領域的基本事實。

要完全理解這本書，讀者需要修讀本科微積分和線性代數課程。我們主要嘗試使用工程直覺而不是堅持數學技巧。這本書的主要特點是其清晰度、動機和作者們對於使描述和組織來自無窮複雜的自然的各種印象的“神秘”數學工具變得極其透明的真誠渴望。

這本書不僅推薦給工程和物理學專業的學生或研究人員，也推薦給數學初學者，因為它展示了純數學與物理現象之間的聯繫，這些聯繫總是為數學發現提供動機。

Sever Angel Popescu. His scientific activity splits into two directions. One is Pure Mathematics, namely, in Algebra and Number theory, where he succeeded in obtaining some interesting results published by known mathematical journals such as Journal of Algebra, Mathematishe Zeitschrift, Mathematiche Nachrihten, Proc. of the Japan Academy, Manuscripta Mathematica, and Acta Arithmetica. (see Google Scholars for instance). The other is in Applied Mathematics, connected to his whole teaching activity at the Technical University of Civil Engineering Bucharest. Here, he tried to apply his mathematical experience to improve the syllabus of advanced mathematics for engineering students. This book covers all his experience in this direction. It is the last book in a series of many other books published in Romanian or in English as courses for engineering students (in Linear Algebra, Mathematical Analysis, Differential Equations, Calculus of Variations, Statistics, etc.). He also published some papers on applied mathematics topics such as differential equations, shift operators, the bridge and torch problem, etc.

Marilena Jianu received her M.Sc. degree in mathematics (statistics and probability) from the University of Bucharest, and her Ph.D. degree in civil engineering from the Technical University of Civil Engineering Bucharest. She has been teaching advanced mathematics to engineering students for over 20 years and has published several papers in pure and applied mathematics. She is an editorial board member of the Romanian Journal of Mathematics and Computer Science and a reviewer for Mathematical Reviews.

Sever Angel Popescu（塞弗·安吉爾·波普斯庫）是一位科學家，他的科學活動分為兩個方向。一個是純數學，特別是代數和數論，他在這方面取得了一些有趣的結果，並發表在著名的數學期刊上，如《Journal of Algebra》、《Mathematishe Zeitschrift》、《Mathematiche Nachrihten》、《Proc. of the Japan Academy》、《Manuscripta Mathematica》和《Acta Arithmetica》（例如，請參閱Google學者）。另一個方向是應用數學，與他在布加勒斯特市土木工程技術大學的整個教學活動相關。在這裡，他試圖應用他的數學經驗來改進工程學生的高級數學課程大綱。這本書涵蓋了他在這個方向上的所有經驗。它是一系列其他許多書籍的最後一本，這些書籍以羅馬尼亞語或英語出版，作為工程學生的課程（如線性代數、數學分析、微分方程、變分計算、統計等）。他還發表了一些應用數學主題的論文，如微分方程、移位運算符、橋和火把問題等。

Marilena Jianu（瑪麗萊娜·吉亞努）在布加勒斯特大學獲得了數學（統計和概率）的碩士學位，並在布加勒斯特市土木工程技術大學獲得了土木工程的博士學位。她已經教授工程學生高級數學超過20年，並在純數學和應用數學方面發表了幾篇論文。她是《羅馬尼亞數學與計算機科學雜誌》的編輯委員會成員，也是《數學評論》的審稿人。