This book presents a history of differential equations, both ordinary
and partial, as well as the calculus of variations, from the origins of
the subjects to around 1900. Topics treated include the wave equation in
the hands of d'Alembert and Euler; Fourier's solutions to the heat
equation and the contribution of Kovalevskaya; the work of Euler, Gauss,
Kummer, Riemann, and Poincaré on the hypergeometric equation; Green's
functions, the Dirichlet principle, and Schwarz's solution of the
Dirichlet problem; minimal surfaces; the telegraphists' equation and
Thomson's successful design of the trans-Atlantic cable; Riemann's paper
on shock waves; the geometrical interpretation of mechanics; and aspects
of the study of the calculus of variations from the problems of the
catenary and the brachistochrone to attempts at a rigorous theory by
Weierstrass, Kneser, and Hilbert. Three final chapters look at how the
theory of partial differential equations stood around 1900, as they were
treated by Picard and Hadamard. There are also extensive, new
translations of original papers by Cauchy, Riemann, Schwarz, Darboux,
and Picard.
The first book to cover the history of differential equations and the
calculus of variations in such breadth and detail, it will appeal to
anyone with an interest in the field. Beyond secondary school
mathematics and physics, a course in mathematical analysis is the only
prerequisite to fully appreciate its contents. Based on a course for
third-year university students, the book contains numerous historical
and mathematical exercises, offers extensive advice to the student on
how to write essays, and can easily be used in whole or in part as a
course in the history of mathematics. Several appendices help make the
book self-contained and suitable for self-study.
