Mathematical Tools for Real-World Applications: A Gentle Introduction for Students and Practitioners

Techniques for applying mathematical concepts in the real world: six rarely taught but crucial tools for analysis, research, and problem-solving.

Many young graduates leave school with a solid knowledge of mathematical concepts but struggle to apply these concepts in practice. Real scientific and engineering problems are different from those found in textbooks: they are messier, take longer to solve, and standard solution recipes might not apply. This book fills the gap between what is taught in the typical college curriculum and what a practicing engineer or scientist needs to know. It presents six powerful tools for analysis, research, and problem-solving in the real world: dimensional analysis, limiting cases, symmetry, scaling, making order of magnitude estimates, and the method of successive approximations.

The book does not focus on formulaic manipulations of equations, but emphasizes analysis and explores connections between the equations and the application. Each chapter introduces a set of ideas and techniques and then shows how these techniques apply to a series of problems. (Knowledge of algebra and trigonometry, but not calculus, is required.) The final two chapters tie all six techniques together and apply them to two real-world problems: computing the probability of a rare, catastrophic event, and tracking a satellite with a GPS receiver. Readers will learn how to analyze, dissect, and gain insight into the results by using all the techniques presented in earlier chapters--and discover how analysis tools work on problems not concocted for a textbook. The appendix provides solutions to many of the problems found throughout the book.

Alexandr Draganov was born and raised in Kyiv, Ukraine; in light of the current war in Ukraine he will donate 100% of his royalties for the first year to support medical and humanitarian efforts there.​

應用數學概念於現實世界的技巧：六種鮮少教授但在分析、研究和解決問題上至關重要的工具。

許多年輕畢業生在學校學習到堅實的數學概念，但在實際應用上卻遇到困難。真實的科學和工程問題與教科書中的問題不同：它們更加複雜，解決時間更長，標準的解決方法可能不適用。本書填補了典型大學課程所教授的內容與實際工程師或科學家所需知識之間的差距。它介紹了六種在現實世界中進行分析、研究和解決問題的強大工具：量綱分析、極限情況、對稱性、尺度變換、量級估計和逐次逼近法。

本書不僅著重於方程式的公式操作，更強調分析並探索方程式與應用之間的聯繫。每個章節介紹一組思想和技巧，然後展示這些技巧如何應用於一系列問題中（需要代數和三角學的知識，但不需要微積分）。最後兩章將六種技巧結合起來，並應用於兩個現實世界的問題：計算罕見災難事件的概率，以及使用GPS接收器追蹤衛星。讀者將學習如何通過使用前幾章介紹的所有技巧來分析、解剖和獲得對結果的洞察力，並發現這些分析工具如何應用於非教科書編造的問題。附錄提供了本書中許多問題的解答。

亞歷山大·德拉加諾夫（Alexandr Draganov）出生並成長於烏克蘭基輔，在烏克蘭目前的戰爭背景下，他將在第一年將100％的版稅捐贈以支持當地的醫療和人道主義工作。

Alexandr Draganov has more than twenty-five years of experience doing research that spans space science (as a graduate student at Stanford) and navigation applications (as a Technical Fellow at Boeing).

亞歷山大·德拉加諾夫擁有超過二十五年的研究經驗，涵蓋太空科學（作為斯坦福大學的研究生）和導航應用（作為波音公司的技術專家）。