Lecture Notes on the Principles and Methods of Applied Mathematics

This textbook aims to equip readers with a deep understanding of mathematical techniques essential for modeling, analyzing, and solving real-world problems across diverse disciplines. Written for graduate students and professionals, the book emphasizes practical applications of applied mathematics in the context of modern challenges, especially in the age of artificial intelligence and data-driven sciences.The text is structured around core areas including complex analysis, differential equations, variational calculus, optimal control, stochastic processes, statistical inference and learning. These foundational topics are developed through a balance of theoretical principles and practical methods, with examples drawn from physics, engineering, and data science to illustrate each technique’s relevance and application. Throughout the book, exercises are proposed to help readers practice and refine these techniques, and the appendices include a collection of past midterm and final exam papers from the University of Arizona’s Math 581 course, offering students a valuable resource for further study and self-assessment.