Mathematical Methods for Physics and Engineering

This book provides an accessible introduction to mathematical methods essential for physics, engineering, and modern computational analysis. Starting from foundational topics such as ordinary and partial differential equations, readers are then introduced to powerful techniques including Fourier and Laplace transforms, series expansions, matrix and eigenvalue methods, and numerical strategies such as iterative refinement.

“Mathematical Methods for Physics and Engineering: Practical Applications” emphasizes intuitive understanding and real-world applications: why the Lagrangian in classical mechanics takes the form T-V; how stability and sensitivity analysis connect to condition numbers and perturbation theory; and how matrix representations provide insight into optimisation and numerical stability.

Numerical examples and step-by-step derivations encourage active problem-solving and demonstrate how abstract methods translate into practical computations. It also highlights how these mathematical tools form the foundation of many techniques used in contemporary machine learning; from optimization algorithms and least-squares regression to spectral methods, kernel functions, and high-dimensional data analysis.

This is an ideal textbook for advanced undergraduate and graduate students studying mathematical methods for physics and/or engineering. Readers are equipped not only a versatile toolkit of methods, but also a deeper conceptual understanding of when, where, and why each tool is appropriate - empowering them to approach problems in physics and engineering.

Key features:

• Provides a toolkit of mathematical methods
• Pedagogically focused, with homework problems included with each chapter
• Covers exciting topics including high-dimensional data analysis and machine learning

Chong Qi is an Associate Professor in the Division of Nuclear Science and Engineering, Department of Physics, at the KTH Royal Institute of Technology, Stockholm. His research spans theoretical nuclear physics, many-body physics, and computational physics. He teaches courses in mathematical physics, subatomic and nuclear physics, and quantum many-body theory, and supervises students at both undergraduate and graduate levels in nuclear physics and nuclear engineering. He earned his PhD from Peking University, followed by postdoctoral research at KTH, and has since held visiting professorships at several universities around the world, including Sun Yat-sen University, where a large part of this book was written. He also serves in various editorial roles and is an honorary member of the Romanian Academy of Scientists.

Chong Qi is an Associate Professor in the Division of Nuclear Science and Engineering, Department of Physics, at the KTH Royal Institute of Technology, Stockholm. His research spans theoretical nuclear physics, many-body physics, and computational physics. He teaches courses in mathematical physics, subatomic and nuclear physics, and quantum many-body theory, and supervises students at both undergraduate and graduate levels in nuclear physics and nuclear engineering. He earned his PhD from Peking University, followed by postdoctoral research at KTH, and has since held visiting professorships at several universities around the world, including Sun Yat-sen University, where a large part of this book was written. He also serves in various editorial roles and is an honorary member of the Romanian Academy of Scientists.