Calculus of Variations

Gilbert Ames Bliss

dec. 1925 · The Carus Mathematical Monographs Bog 1 · American Mathematical Soc.

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The development of the calculus of variations has, from the beginning, been interlaced with that of the differential and integral calculus. Without any knowledge of the calculus, one can readily understand at least the geometrical or mechanical statements of many of the problems of the calculus of variations and the character of their solutions. The discovery and justification of the results in this book, apart from their simple statements, do require, however, acquaintance with the principles of the calculus, and it is assumed that the reader has such an acquaintance. Calculus of Variations begins by studying special problems rather than the general theory.

The first chapter of the book describes the historical setting out of which the theory of the calculus of variations grew and the character of some of the simpler problems. The next three chapters are devoted to the development, in detail, of the then known results for three special problems (shortest distances, brachistochrone, and surfaces of revolution of minimum area) which illustrate in excellent fashion the essential characteristics of the general theory contained in Chapter V with which the book concludes.

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