Lectures on the Calculus of Variations

This pioneering modern treatise on the calculus of variations studies the evolution of the subject from Euler to Hilbert. The text addresses basic problems with sufficient generality and rigor to offer a sound introduction for serious study. It provides clear definitions of the fundamental concepts, sharp formulations of the problems, and rigorous demonstrations of their solutions. Many examples are solved completely, and systematic references are given for each theorem upon its first appearance.
Initial chapters address the first and second variation of the integral, and succeeding chapters cover the sufficient conditions for an extremum of the integral and Weierstrass's theory of the problem in parameter-representation; Kneser's extension of Weierstrass's theory to cover the case of variable end-points; and Weierstrass's theory of the isoperimetric problems. The final chapter presents a thorough proof of Hilbert's existence theorem.

Oskar Bolza (1857–1942) studied mathematics at the University of Berlin and later at Strasbourg. He taught at Johns Hopkins, Clark University, and the University of Chicago, returning to Germany 22 years later but retaining the University of Chicago title of Non-resident Professor of Mathematics for the rest of his life. He taught mathematics at the University of Freiburg until his 1933 retirement.

I. The First Variation of the IntegralII. The Second Variation of the IntegralIII. Sufficient Conditions for an Extremum of the IntegralIV. Weierstrass's Theory of the Problem in Parameter-RepresentationV. Kneser's TheoryVI. Weierstrass's Theory of the Isoperimetric ProblemsVII. Hilbert's Existence TheoremAddendaIndex