Definitely, Mathematical Methods in Science and Engineering, second edition, is a textbook providing numerous mathematical techniques and tools that may be applied to different scientific and engineering disciplines. Most of these applied mathematical methods have been used particularly in theoretical physics. Nevertheless, the idea of the new edition is to expand the applications beyond physics, to numerous other scientific disciplines such as chemistry, biology, economy, and finance, showing the utility of these advanced mathematical methods. The book is organised into 19 chapters, where the different mathematical methods are included. Even though the book lacks sections or parts, the first eight chapters are devoted to special functions, such as Legendre, Laguerre, Hermite, and Chebyschev polynomials, including special functions such as Bessel andHypergeometric functions, and the Sturm-Liouville theory. The structure of each chapter is rather similar, and contains a detailed description of the subject matter with applications in physics and other disciplines. Some interesting particular examples of applications are included, and at the end of each chapter there is a brief and useful bibliography on the topic and its applications, and a collection of unsolved problems. Among the applications, there are examples in classical and quantum physics, and computer graphics. An interesting and rather comprehensive chapter focuses on coordinate systems, vectors, tensors, and generalised coordinateswith special applications to special relativity and the theory of elasticity. Besides being self-contained, it can be used independently from the rest of the book, providing an excellent overview of its applicability to science and engineering, and in this case to physics. Group theory, including Lie groups and Lie algebras, is contained in another chapter, considering ideas of symmetries in whatever discipline in physics. It can be useful to refresh the ideas for those who already know, or to learn them from those who do not know, in any case in a synthetic manner. A couple of chapters consider complex variables and integrals, with numerous applications to electrostatics and fluids, where some special functions such as Gamma and Beta functions are included. Fractional calculus, its mathematical techniques and applications to science and engineering constitute another chapter. Much research is currently underway using techniques from fractional differential equations so that this chapter is a good asset for that purpose. Another group of chapters includes infinite series, Fourier and Laplace transformations and integral transformations. Variational analysis and applications to classical mechanics and Hamiltonian equations and Hamilton principle with applications to control and dynamics are the subjects of another chapter. Finally, one chapter covers integral equations with applications, and Green’s functions, path integrals and path integral Feynman formulation of quantum mechanics are offered in the final chapters. Precisely in relation to mathematical methods for the sciences, it is a pity that the terminology and the nomenclature are not unified, since this can contribute to a certain confusion. Traditionally this has been the case with certain topics such as vectors and tensors, as well as others, which receive different notations in areas of physics, engineering and applied mathematics. A greater unification would be important, since fortunately, as this textbook shows, themethods are of general application to the sciences and engineering. It would have been desirable that new references would have been added to the bibliography in the second edition. Another aspect that would have much improved the book would have been to classify the different techniques and methods in sections or parts. Furthermore, in spite of the importance and relevance that nowadays have the computational methods and numerical algorithms, these are lacking in the textbook. The text can be used for the classroom, as well as a reference for students and researchers. As I commented earlier, many chapters or groups of chapters can be read independently from one another, so it can be used also for selfstudy. In any case, it gives the readers the strong foundation needed to apply mathematical methods to the physical phenomena found in scientific and engineering applications. From this perspective, the book can be of interest to a wide audience, ranging from postgraduates and researchers in physics, engineering and other scientific disciplines interested in advanced mathematical methods of analysis.

中文翻译：

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科学与工程中的数学方法，第 2 版

毫无疑问，《科学与工程中的数学方法》第二版是一本教科书，提供了许多可应用于不同科学和工程学科的数学技术和工具。这些应用数学方法中的大多数已特别用于理论物理学。尽管如此，新版本的想法是将应用扩展到物理之外，扩展到许多其他科学学科，如化学、生物学、经济和金融，展示这些先进数学方法的实用性。本书分为 19 章，其中包括不同的数学方法。尽管本书缺少章节或部分，但前八章专门介绍了特殊函数，例如勒让德、拉盖尔、埃尔米特和切比雪夫多项式，包括特殊函数，如贝塞尔和超几何函数，以及 Sturm-Liouville 理论。每一章的结构都相当相似，并包含对该主题的详细描述，以及在物理学和其他学科中的应用。包括一些有趣的特定应用示例，每章末尾都有关于该主题及其应用的简短而有用的参考书目，以及一系列未解决的问题。在这些应用中，有经典物理学和量子物理学以及计算机图形学方面的例子。一个有趣且相当全面的章节侧重于坐标系、向量、张量和广义坐标以及对狭义相对论和弹性理论的特殊应用。除了自给自足，它可以独立于本书的其余部分使用，很好地概述了它在科学和工程中的适用性，在这种情况下也适用于物理学。群论，包括李群和李代数，包含在另一章中，考虑了物理学中任何学科的对称性思想。在任何情况下以综合的方式为那些已经知道的人更新想法，或者向不知道的人学习它们是有用的。有几章考虑了复变量和积分，以及对静电和流体的大量应用，其中包括了一些特殊函数，例如 Gamma 和 Beta 函数。分数阶微积分、它的数学技术和在科学和工程中的应用构成了另一章。目前正在使用分数阶微分方程的技术进行大量研究，因此本章是实现这一目的的好资产。另一组章节包括无限级数、傅立叶和拉普拉斯变换以及积分变换。变分分析和经典力学和哈密顿方程的应用和哈密顿原理以及控制和动力学的应用是另一章的主题。最后一章介绍了积分方程及其应用，最后几章提供了量子力学的格林函数、路径积分和路径积分费曼公式。确切地说，就科学的数学方法而言，遗憾的是术语和命名法没有统一，因为这可能会导致某种混淆。传统上，某些主题就是这种情况，例如向量和张量，以及其他在物理、工程和应用数学领域接受不同符号的主题。更大的统一将很重要，因为幸运的是，正如这本教科书所示，这些方法普遍适用于科学和工程。在第二版的参考书目中添加新的参考文献是可取的。另一个可以大大改进本书的方面是将不同的技术和方法分类为部分或部分。此外，尽管如今计算方法和数值算法具有重要性和相关性，但这些在教科书中是缺乏的。课文可用于课堂，以及供学生和研究人员参考。正如我之前评论过的，许多章节或章节组可以相互独立阅读，因此也可以用于自学。无论如何，它为读者提供了将数学方法应用于科学和工程应用中发现的物理现象所需的坚实基础。从这个角度来看，本书可能会引起广大读者的兴趣，包括对高级数学分析方法感兴趣的物理学、工程和其他科学学科的研究生和研究人员。它为读者提供了将数学方法应用于科学和工程应用中的物理现象所需的坚实基础。从这个角度来看，本书可能会引起广大读者的兴趣，包括对高级数学分析方法感兴趣的物理学、工程和其他科学学科的研究生和研究人员。它为读者提供了将数学方法应用于科学和工程应用中的物理现象所需的坚实基础。从这个角度来看，本书可能会引起广大读者的兴趣，包括对高级数学分析方法感兴趣的物理学、工程和其他科学学科的研究生和研究人员。