Kármán v. T., Biot M.A. Mathematical Methods in Engineering. An Introduction to the Mathematical Treatment of Engineering Problems

New York: McGraw-Hill. – 1940. – 517 p. The primary objective of this book is to introduce the reader to the mathematical treatment of engineering problems. It is often said that the engineer nowadays needs more and more mathematics. However, it is common experience that most, engineers apply only a small portion of the knowledge presented to them in their college mathematics courses. It seems that it many cases the amount of mathematics included in the curriculums is quite adequate, but the ability to find the proper mathematical setup for given physical or engineering problems is not developed in the student to a sufficient degree. In other words, the need is not so much for more mathematics as for a better understanding of the potentialities of its application. There are two ways of teaching the art of applying mathematics to engineering problems. One consists of a systematic course comprising selected branches of mathematics including a choice of appropriate examples for applications. The other chooses certain representative groups of engineering problems and demonstrates the mathematical approach to their solution. There are excellent books available that follow the first method. This book might be considered as an experiment in the direction of the second method. Of course, it would not be advantageous to push this method to an extreme; hence, certain parts of' the book are concerned primarily with mathematical subjects. However, also in these parts the mathematical concepts are presented from the viewpoint of their application rather than from that of their purely logical development. The first two chapters on differential equations and Bessel functions are fundamentally of a mathematical character. In the subsequent chapters we have tried to insert most of the information of mathematical nature between the problems that require their knowledge. The reader will notice that elliptic integrals, for example, are treated in connection with the classical problem of the pendulum, and that vector algebra appears in the chapter on the fundamental concepts of dynamics. The latter chapter was included in the belief that mechanica rationalis generally does not enjoy its due place between physics and applied mechanics in the engineering curriculum. The instructor who shuns Lagrange's equations may omit the last sections of Chapter III. The examples of Chapters V and VI, in which Lagrange’s equations are used, can be treated without reference to the Lagrangian method. However, it is believed that familiarity with generalized coordinates, generalized forces, and other concepts of the mechanics of Lagrange, is a positive advantage for the scientific minded engineer.Introduction to ordinary differential equations
Some information on Bessel functions
Fundamental concepts of dynamics
Elementary problems in dynamics
Small oscillations of conservative systems
Small oscillations of nonconservative systems
The differential equations of the theory of structures
Fourier series applied to structural problems
Complex representation of periodic phenomena
Transient phenomena. Operational calculus
Equations with finite differences applied to engineering
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