Lectures on the Calculus of Variations (Dover Books on Mathematics)
(This pioneering modern treatise on the calculus of variat...)
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.
Mathematical papers read at the International Mathematical Congress held in connection with the Worl
(This is a pre-1923 historical reproduction that was curat...)
This is a pre-1923 historical reproduction that was curated for quality. Quality assurance was conducted on each of these books in an attempt to remove books with imperfections introduced by the digitization process. Though we have made best efforts - the books may have occasional errors that do not impede the reading experience. We believe this work is culturally important and have elected to bring the book back into print as part of our continuing commitment to the preservation of printed works worldwide. This text refers to the Bibliobazaar edition.
Oskar Bolza was a German mathematician. He was Nonresident Professor of Mathematics at the University of Chicago.
Background
Oskar Bolza was born on May 12, 1857, in Bad Bergzabern, Germany, the oldest of four children of Moritz and Luise (Koenig) Bolza. His maternal grandfather was Friedrich Koenig, whose fortune derived from the development and manufacture of a rapid printing press. His father was in the judicial service, and during Bolza's early life the family moved from place to place in southern Germany until 1873, when they settled at Freiburg im Breisgau.
Education
It was hoped that Oskar Bolza would enter the family printing press factory, and to this end he studied for a time at a Gewerbeakademie while enrolled at the University of Berlin. His scholarly bent, however, won out. At first interested in languages and comparative philology, he was soon drawn to science. He initially tried physics, but experimental work did not attract him and in 1878 he decided to concentrate on mathematics. Bolza spent the years 1878-1881 in mathematical study under Elwin B. Christoffel and Theodor Reye at Strassburg, Hermann A. Schwarz at Göttingen, and particularly Karl Weierstrass at the University of Berlin. Undoubtedly, the fact that he was a student in the famous 1879 course of Weierstrass on the calculus of variations exerted a strong influence on the formation of Bolza's mathematical interests. After preparing for and passing the Staatsexamen Bolza in 1883 returned to mathematical study and undertook a doctoral dissertation on the determination of hyperelliptic integrals which are reducible to elliptic integrals by a transformation of the third degree. He obtained a solution to his problem in 1885, but was anticipated by a more elegant one published by Ïdouard Goursat in the Comptes Rendus. Bolsa then turned to the corresponding problem for transformations of the fourth degree. His solution, and subsequent development of the work, formed the basis for his doctoral dissertation, under Felix Klein, and he received his degree from the University of Göttingen in June 1886.
Career
As a student in Berlin, Bolza had made two intimate friends, the mathematician Heinrich Maschke and the physicist Franz Schulze-Berge. Schulze-Berge migrated to the United States in 1887, and his experience stimulated Bolza to follow a year later. Bolza secured a post as reader in mathematics at Johns Hopkins University in January 1889, and in October was appointed associate in mathematics at Clark University. In 1892 he was invited to join the faculty of the newly established University of Chicago, where he began teaching in January 1893. He persuaded Eliakim H. Moore, the head of the department of mathematics, to secure an appointment for his friend Maschke as well. The three - Moore, Bolza, and Maschke - provided great strength for the department, and the Chicago graduate school of mathematics became at once one of the leaders in this country.
In the years following, Bolza took an active part in the International Mathematical Congress held at the Chicago World's Fair of 1893, in the colloquium lectures given afterward by Klein at Northwestern University, and in the transformation (1894) of the New York Mathematical Society into the American Mathematical Society, which he served as vice-president in 1903-1904. At its third colloquium, in Ithaca in August 1901, he presented a series of lectures on the calculus of variations which were published in book form in 1904. Thereafter Bolza's primary mathematical interests centered in this subject. In 1908-1909 Bolza published a much enlarged treatise, Vorlesungen über Variationsrechnung, which sixty years later remained a classic in its field. During his years at Chicago nine students wrote doctoral dissertations under Bolza's guidance: three in the area of hyperelliptic integrals, one on singularities of algebraic curves, and five on variational theory. Of the last group, one was Gilbert A. Bliss (1900), who was a member of the mathematics faculty of the University of Chicago from 1908 to 1941, and who deserves major credit for the continuation of a strong school in the calculus of variations.
Through the years Bolza made many trips back to Freiburg, where his mother continued to live after the death of his father in 1891. Maschke died in 1908, and two years later Bolza decided to return to Freiburg; before he left, the University of Chicago conferred on him the title of "Nonresident Professor of Mathematics, " which he held until his death. For several years Bolza continued his mathematical lectures and research, as an honorary professor at the University of Freiburg. He returned to the United States for the last time in 1913 and lectured during the summer at the University of Chicago on function theory and integral equations. One of his students, William V. Lovitt, later published, with Bolza's permission, the notes he had taken on these lectures as Linear Integral Equations (1924).
The First World War put an end to Bolza's mathematical research. He continued to lecture until 1926, and again from 1929 to 1933, but from 1917 onward his principal interests were religious psychology and languages, particularly Sanskrit. In 1930 he published Glaubenslose Religion ("Religion without Belief"), under the pseudonym F. N. Marneck, and in 1938 a short monograph, Meister Eckehart als Mystiker, a study of a fourteenth-century German Dominican friar and mystic. The death of his wife in 1941 was a severe blow, and after a year of progressively failing health, Bolza died in Freiburg at the age of eighty-five.